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Properties

Label 2-2548-91.88-c1-0-27
Degree 22
Conductor 25482548
Sign 0.845−0.533i0.845 - 0.533i
Analytic cond. 20.345820.3458
Root an. cond. 4.510644.51064
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  + 2.98·3-s + (0.626 − 0.361i)5-s + 5.89·9-s + 2.58i·11-s + (0.896 + 3.49i)13-s + (1.86 − 1.07i)15-s + (−2.26 − 3.93i)17-s + 7.22i·19-s + (−0.0997 + 0.172i)23-s + (−2.23 + 3.87i)25-s + 8.64·27-s + (2.84 + 4.93i)29-s + (5.36 + 3.09i)31-s + 7.70i·33-s + (−4.01 − 2.32i)37-s + ⋯
L(s)  = 1  + 1.72·3-s + (0.280 − 0.161i)5-s + 1.96·9-s + 0.779i·11-s + (0.248 + 0.968i)13-s + (0.482 − 0.278i)15-s + (−0.550 − 0.953i)17-s + 1.65i·19-s + (−0.0208 + 0.0360i)23-s + (−0.447 + 0.775i)25-s + 1.66·27-s + (0.529 + 0.916i)29-s + (0.963 + 0.556i)31-s + 1.34i·33-s + (−0.660 − 0.381i)37-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((0.845−0.533i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s+1/2) L(s)=((0.845−0.533i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: 0.845−0.533i0.845 - 0.533i
Analytic conductor: 20.345820.3458
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2548(361,⋅)\chi_{2548} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :1/2), 0.845−0.533i)(2,\ 2548,\ (\ :1/2),\ 0.845 - 0.533i)

Particular Values

L(1)L(1) ≈\approx 3.6607771353.660777135
L(12)L(\frac12) ≈\approx 3.6607771353.660777135
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 1
7 1 1
13 1+(−0.896−3.49i)T 1 + (-0.896 - 3.49i)T
good3 1−2.98T+3T2 1 - 2.98T + 3T^{2}
5 1+(−0.626+0.361i)T+(2.5−4.33i)T2 1 + (-0.626 + 0.361i)T + (2.5 - 4.33i)T^{2}
11 1−2.58iT−11T2 1 - 2.58iT - 11T^{2}
17 1+(2.26+3.93i)T+(−8.5+14.7i)T2 1 + (2.26 + 3.93i)T + (-8.5 + 14.7i)T^{2}
19 1−7.22iT−19T2 1 - 7.22iT - 19T^{2}
23 1+(0.0997−0.172i)T+(−11.5−19.9i)T2 1 + (0.0997 - 0.172i)T + (-11.5 - 19.9i)T^{2}
29 1+(−2.84−4.93i)T+(−14.5+25.1i)T2 1 + (-2.84 - 4.93i)T + (-14.5 + 25.1i)T^{2}
31 1+(−5.36−3.09i)T+(15.5+26.8i)T2 1 + (-5.36 - 3.09i)T + (15.5 + 26.8i)T^{2}
37 1+(4.01+2.32i)T+(18.5+32.0i)T2 1 + (4.01 + 2.32i)T + (18.5 + 32.0i)T^{2}
41 1+(−8.26+4.77i)T+(20.5−35.5i)T2 1 + (-8.26 + 4.77i)T + (20.5 - 35.5i)T^{2}
43 1+(−3.54+6.14i)T+(−21.5−37.2i)T2 1 + (-3.54 + 6.14i)T + (-21.5 - 37.2i)T^{2}
47 1+(−0.838+0.484i)T+(23.5−40.7i)T2 1 + (-0.838 + 0.484i)T + (23.5 - 40.7i)T^{2}
53 1+(−2.13+3.69i)T+(−26.5−45.8i)T2 1 + (-2.13 + 3.69i)T + (-26.5 - 45.8i)T^{2}
59 1+(−7.48+4.32i)T+(29.5−51.0i)T2 1 + (-7.48 + 4.32i)T + (29.5 - 51.0i)T^{2}
61 1+8.73T+61T2 1 + 8.73T + 61T^{2}
67 1+2.73iT−67T2 1 + 2.73iT - 67T^{2}
71 1+(−0.0784−0.0453i)T+(35.5+61.4i)T2 1 + (-0.0784 - 0.0453i)T + (35.5 + 61.4i)T^{2}
73 1+(3.56+2.05i)T+(36.5+63.2i)T2 1 + (3.56 + 2.05i)T + (36.5 + 63.2i)T^{2}
79 1+(3.96+6.86i)T+(−39.5+68.4i)T2 1 + (3.96 + 6.86i)T + (-39.5 + 68.4i)T^{2}
83 1+6.65iT−83T2 1 + 6.65iT - 83T^{2}
89 1+(12.2+7.06i)T+(44.5+77.0i)T2 1 + (12.2 + 7.06i)T + (44.5 + 77.0i)T^{2}
97 1+(−1.09−0.630i)T+(48.5+84.0i)T2 1 + (-1.09 - 0.630i)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

−8.838261380483597109727043110044, −8.472497042167133697265518993057, −7.33810788886002947577920602789, −7.13975639960289922225137269480, −5.92346157382717000358004135428, −4.75635101646456839669547996740, −4.01547463391556352978777853192, −3.22014007626498053791280722523, −2.17470728437372467708357205802, −1.57257452655174935348314333156, 1.02535750293550658023507396986, 2.55573076290977351127214317372, 2.72862689368366647366409812724, 3.88209192911069144222479127612, 4.56389463314289890341200807458, 5.90189019590505853363567565196, 6.57985195486832597177844471683, 7.62243660169443712818100252285, 8.217605902221525229591244669191, 8.684137292843395640364953821164

Graph of the ZZ-function along the critical line