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Properties

Label 2-1098-9.4-c1-0-55
Degree 22
Conductor 10981098
Sign −0.250−0.968i-0.250 - 0.968i
Analytic cond. 8.767578.76757
Root an. cond. 2.961002.96100
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  + (−0.5 − 0.866i)2-s + (−1.53 − 0.806i)3-s + (−0.499 + 0.866i)4-s + (1.59 − 2.76i)5-s + (0.0679 + 1.73i)6-s + (−2.01 − 3.48i)7-s + 0.999·8-s + (1.69 + 2.47i)9-s − 3.19·10-s + (−2.90 − 5.03i)11-s + (1.46 − 0.924i)12-s + (−2.04 + 3.54i)13-s + (−2.01 + 3.48i)14-s + (−4.67 + 2.95i)15-s + (−0.5 − 0.866i)16-s − 3.84·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.884 − 0.465i)3-s + (−0.249 + 0.433i)4-s + (0.713 − 1.23i)5-s + (0.0277 + 0.706i)6-s + (−0.761 − 1.31i)7-s + 0.353·8-s + (0.566 + 0.824i)9-s − 1.00·10-s + (−0.877 − 1.51i)11-s + (0.422 − 0.266i)12-s + (−0.566 + 0.982i)13-s + (−0.538 + 0.932i)14-s + (−1.20 + 0.761i)15-s + (−0.125 − 0.216i)16-s − 0.932·17-s + ⋯

Functional equation

Λ(s)=(1098s/2 ΓC(s) L(s)=((−0.250−0.968i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1098s/2 ΓC(s+1/2) L(s)=((−0.250−0.968i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.250 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10981098    =    2⋅32⋅612 \cdot 3^{2} \cdot 61
Sign: −0.250−0.968i-0.250 - 0.968i
Analytic conductor: 8.767578.76757
Root analytic conductor: 2.961002.96100
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1098(733,⋅)\chi_{1098} (733, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1098, ( :1/2), −0.250−0.968i)(2,\ 1098,\ (\ :1/2),\ -0.250 - 0.968i)

Particular Values

L(1)L(1) ≈\approx 0.51189728140.5118972814
L(12)L(\frac12) ≈\approx 0.51189728140.5118972814
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
3 1+(1.53+0.806i)T 1 + (1.53 + 0.806i)T
61 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good5 1+(−1.59+2.76i)T+(−2.5−4.33i)T2 1 + (-1.59 + 2.76i)T + (-2.5 - 4.33i)T^{2}
7 1+(2.01+3.48i)T+(−3.5+6.06i)T2 1 + (2.01 + 3.48i)T + (-3.5 + 6.06i)T^{2}
11 1+(2.90+5.03i)T+(−5.5+9.52i)T2 1 + (2.90 + 5.03i)T + (-5.5 + 9.52i)T^{2}
13 1+(2.04−3.54i)T+(−6.5−11.2i)T2 1 + (2.04 - 3.54i)T + (-6.5 - 11.2i)T^{2}
17 1+3.84T+17T2 1 + 3.84T + 17T^{2}
19 1−3.74T+19T2 1 - 3.74T + 19T^{2}
23 1+(−3.09+5.35i)T+(−11.5−19.9i)T2 1 + (-3.09 + 5.35i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.82+3.16i)T+(−14.5+25.1i)T2 1 + (1.82 + 3.16i)T + (-14.5 + 25.1i)T^{2}
31 1+(2.97−5.15i)T+(−15.5−26.8i)T2 1 + (2.97 - 5.15i)T + (-15.5 - 26.8i)T^{2}
37 1−7.36T+37T2 1 - 7.36T + 37T^{2}
41 1+(−2.02+3.51i)T+(−20.5−35.5i)T2 1 + (-2.02 + 3.51i)T + (-20.5 - 35.5i)T^{2}
43 1+(5.31+9.20i)T+(−21.5+37.2i)T2 1 + (5.31 + 9.20i)T + (-21.5 + 37.2i)T^{2}
47 1+(−4.78−8.29i)T+(−23.5+40.7i)T2 1 + (-4.78 - 8.29i)T + (-23.5 + 40.7i)T^{2}
53 1−6.91T+53T2 1 - 6.91T + 53T^{2}
59 1+(4.60−7.97i)T+(−29.5−51.0i)T2 1 + (4.60 - 7.97i)T + (-29.5 - 51.0i)T^{2}
67 1+(0.494−0.857i)T+(−33.5−58.0i)T2 1 + (0.494 - 0.857i)T + (-33.5 - 58.0i)T^{2}
71 1−6.71T+71T2 1 - 6.71T + 71T^{2}
73 1+15.9T+73T2 1 + 15.9T + 73T^{2}
79 1+(−6.03−10.4i)T+(−39.5+68.4i)T2 1 + (-6.03 - 10.4i)T + (-39.5 + 68.4i)T^{2}
83 1+(0.758+1.31i)T+(−41.5+71.8i)T2 1 + (0.758 + 1.31i)T + (-41.5 + 71.8i)T^{2}
89 1−0.883T+89T2 1 - 0.883T + 89T^{2}
97 1+(−2.85−4.94i)T+(−48.5+84.0i)T2 1 + (-2.85 - 4.94i)T + (-48.5 + 84.0i)T^{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

−9.324548671050771141065698158836, −8.680817695723821590553931174446, −7.57383530520536516156876466759, −6.78554625247136334939360323690, −5.80453278317862720873278572280, −4.91638410824573627313768210879, −4.06323515635081365381489479997, −2.54656313943366307167179383508, −1.13477466060181891643063108610, −0.31952976631360448718259655911, 2.20440622425686146578572097486, 3.15023124305238797847904287884, 4.85167265592084476467171959266, 5.54082600505629664610158316971, 6.16199181465808209992094924342, 7.02703425395360669618675039297, 7.62500751501548542659568987978, 9.206805348150377855300190466991, 9.708200233259377748846739452361, 10.14573521455820717719867220843

Graph of the ZZ-function along the critical line