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Properties

Label 2-2e7-128.109-c1-0-13
Degree 22
Conductor 128128
Sign 0.711+0.702i0.711 + 0.702i
Analytic cond. 1.022081.02208
Root an. cond. 1.010981.01098
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.603 − 1.27i)2-s + (1.98 + 1.06i)3-s + (−1.27 − 1.54i)4-s + (−0.212 + 0.259i)5-s + (2.55 − 1.89i)6-s + (−0.792 − 0.529i)7-s + (−2.74 + 0.694i)8-s + (1.14 + 1.71i)9-s + (0.203 + 0.428i)10-s + (2.34 + 0.709i)11-s + (−0.886 − 4.41i)12-s + (−2.81 + 2.30i)13-s + (−1.15 + 0.693i)14-s + (−0.696 + 0.288i)15-s + (−0.766 + 3.92i)16-s + (−1.82 − 0.756i)17-s + ⋯
L(s)  = 1  + (0.426 − 0.904i)2-s + (1.14 + 0.612i)3-s + (−0.635 − 0.771i)4-s + (−0.0950 + 0.115i)5-s + (1.04 − 0.775i)6-s + (−0.299 − 0.200i)7-s + (−0.969 + 0.245i)8-s + (0.382 + 0.573i)9-s + (0.0641 + 0.135i)10-s + (0.705 + 0.214i)11-s + (−0.255 − 1.27i)12-s + (−0.779 + 0.640i)13-s + (−0.308 + 0.185i)14-s + (−0.179 + 0.0745i)15-s + (−0.191 + 0.981i)16-s + (−0.442 − 0.183i)17-s + ⋯

Functional equation

Λ(s)=(128s/2 ΓC(s) L(s)=((0.711+0.702i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(128s/2 ΓC(s+1/2) L(s)=((0.711+0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 128128    =    272^{7}
Sign: 0.711+0.702i0.711 + 0.702i
Analytic conductor: 1.022081.02208
Root analytic conductor: 1.010981.01098
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ128(109,⋅)\chi_{128} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 128, ( :1/2), 0.711+0.702i)(2,\ 128,\ (\ :1/2),\ 0.711 + 0.702i)

Particular Values

L(1)L(1) ≈\approx 1.45536−0.597542i1.45536 - 0.597542i
L(12)L(\frac12) ≈\approx 1.45536−0.597542i1.45536 - 0.597542i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(−0.603+1.27i)T 1 + (-0.603 + 1.27i)T
good3 1+(−1.98−1.06i)T+(1.66+2.49i)T2 1 + (-1.98 - 1.06i)T + (1.66 + 2.49i)T^{2}
5 1+(0.212−0.259i)T+(−0.975−4.90i)T2 1 + (0.212 - 0.259i)T + (-0.975 - 4.90i)T^{2}
7 1+(0.792+0.529i)T+(2.67+6.46i)T2 1 + (0.792 + 0.529i)T + (2.67 + 6.46i)T^{2}
11 1+(−2.34−0.709i)T+(9.14+6.11i)T2 1 + (-2.34 - 0.709i)T + (9.14 + 6.11i)T^{2}
13 1+(2.81−2.30i)T+(2.53−12.7i)T2 1 + (2.81 - 2.30i)T + (2.53 - 12.7i)T^{2}
17 1+(1.82+0.756i)T+(12.0+12.0i)T2 1 + (1.82 + 0.756i)T + (12.0 + 12.0i)T^{2}
19 1+(−0.157+1.60i)T+(−18.6−3.70i)T2 1 + (-0.157 + 1.60i)T + (-18.6 - 3.70i)T^{2}
23 1+(7.27+1.44i)T+(21.2+8.80i)T2 1 + (7.27 + 1.44i)T + (21.2 + 8.80i)T^{2}
29 1+(−1.10−3.64i)T+(−24.1+16.1i)T2 1 + (-1.10 - 3.64i)T + (-24.1 + 16.1i)T^{2}
31 1+(−7.16+7.16i)T−31iT2 1 + (-7.16 + 7.16i)T - 31iT^{2}
37 1+(−0.968+0.0953i)T+(36.2−7.21i)T2 1 + (-0.968 + 0.0953i)T + (36.2 - 7.21i)T^{2}
41 1+(2.34−11.8i)T+(−37.8−15.6i)T2 1 + (2.34 - 11.8i)T + (-37.8 - 15.6i)T^{2}
43 1+(−7.65+4.09i)T+(23.8−35.7i)T2 1 + (-7.65 + 4.09i)T + (23.8 - 35.7i)T^{2}
47 1+(1.74−4.20i)T+(−33.2−33.2i)T2 1 + (1.74 - 4.20i)T + (-33.2 - 33.2i)T^{2}
53 1+(−2.50+8.26i)T+(−44.0−29.4i)T2 1 + (-2.50 + 8.26i)T + (-44.0 - 29.4i)T^{2}
59 1+(2.07+1.69i)T+(11.5+57.8i)T2 1 + (2.07 + 1.69i)T + (11.5 + 57.8i)T^{2}
61 1+(−3.63+6.80i)T+(−33.8−50.7i)T2 1 + (-3.63 + 6.80i)T + (-33.8 - 50.7i)T^{2}
67 1+(−5.45+10.1i)T+(−37.2−55.7i)T2 1 + (-5.45 + 10.1i)T + (-37.2 - 55.7i)T^{2}
71 1+(−1.79+2.68i)T+(−27.1−65.5i)T2 1 + (-1.79 + 2.68i)T + (-27.1 - 65.5i)T^{2}
73 1+(−2.04+1.36i)T+(27.9−67.4i)T2 1 + (-2.04 + 1.36i)T + (27.9 - 67.4i)T^{2}
79 1+(−4.58−11.0i)T+(−55.8+55.8i)T2 1 + (-4.58 - 11.0i)T + (-55.8 + 55.8i)T^{2}
83 1+(7.11+0.700i)T+(81.4+16.1i)T2 1 + (7.11 + 0.700i)T + (81.4 + 16.1i)T^{2}
89 1+(13.6−2.71i)T+(82.2−34.0i)T2 1 + (13.6 - 2.71i)T + (82.2 - 34.0i)T^{2}
97 1+(10.5−10.5i)T−97iT2 1 + (10.5 - 10.5i)T - 97iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.38060326874019966843054473905, −12.17164575850935274967859368703, −11.22479497095736126491332701514, −9.775346433633423514124290137508, −9.499817961896115880718057333951, −8.247556907801319429593014282121, −6.53154013924849264406527978204, −4.63377673567862642252277302680, −3.68576079423547044827991203592, −2.37054950194748650624241150838, 2.73245226033875988404517243887, 4.16468891146697493318092592732, 5.87984096326876587963749272519, 7.06895096284082919626259469921, 8.098027034895520802424645522991, 8.768277508092425281253949398167, 9.976818165117118138070720093264, 12.04323457346639423123344997273, 12.65812819817824719794984719292, 13.89429350761274487378094730068

Graph of the ZZ-function along the critical line