Properties

Label 2-42e2-9.4-c1-0-20
Degree 22
Conductor 17641764
Sign 0.313+0.949i0.313 + 0.949i
Analytic cond. 14.085614.0856
Root an. cond. 3.753083.75308
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  + (−1.72 + 0.124i)3-s + (−1.73 + 3.01i)5-s + (2.96 − 0.431i)9-s + (−1.25 − 2.17i)11-s + (−0.292 + 0.505i)13-s + (2.62 − 5.42i)15-s + 1.09·17-s − 5.93·19-s + (−3.19 + 5.52i)23-s + (−3.55 − 6.15i)25-s + (−5.07 + 1.11i)27-s + (0.918 + 1.59i)29-s + (−3.51 + 6.09i)31-s + (2.44 + 3.60i)33-s − 1.40·37-s + ⋯
L(s)  = 1  + (−0.997 + 0.0721i)3-s + (−0.778 + 1.34i)5-s + (0.989 − 0.143i)9-s + (−0.379 − 0.656i)11-s + (−0.0810 + 0.140i)13-s + (0.678 − 1.40i)15-s + 0.265·17-s − 1.36·19-s + (−0.665 + 1.15i)23-s + (−0.710 − 1.23i)25-s + (−0.976 + 0.214i)27-s + (0.170 + 0.295i)29-s + (−0.631 + 1.09i)31-s + (0.425 + 0.627i)33-s − 0.231·37-s + ⋯

Functional equation

Λ(s)=(1764s/2ΓC(s)L(s)=((0.313+0.949i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1764s/2ΓC(s+1/2)L(s)=((0.313+0.949i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17641764    =    2232722^{2} \cdot 3^{2} \cdot 7^{2}
Sign: 0.313+0.949i0.313 + 0.949i
Analytic conductor: 14.085614.0856
Root analytic conductor: 3.753083.75308
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1764(589,)\chi_{1764} (589, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1764, ( :1/2), 0.313+0.949i)(2,\ 1764,\ (\ :1/2),\ 0.313 + 0.949i)

Particular Values

L(1)L(1) \approx 0.34765190640.3476519064
L(12)L(\frac12) \approx 0.34765190640.3476519064
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(1.720.124i)T 1 + (1.72 - 0.124i)T
7 1 1
good5 1+(1.733.01i)T+(2.54.33i)T2 1 + (1.73 - 3.01i)T + (-2.5 - 4.33i)T^{2}
11 1+(1.25+2.17i)T+(5.5+9.52i)T2 1 + (1.25 + 2.17i)T + (-5.5 + 9.52i)T^{2}
13 1+(0.2920.505i)T+(6.511.2i)T2 1 + (0.292 - 0.505i)T + (-6.5 - 11.2i)T^{2}
17 11.09T+17T2 1 - 1.09T + 17T^{2}
19 1+5.93T+19T2 1 + 5.93T + 19T^{2}
23 1+(3.195.52i)T+(11.519.9i)T2 1 + (3.19 - 5.52i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.9181.59i)T+(14.5+25.1i)T2 1 + (-0.918 - 1.59i)T + (-14.5 + 25.1i)T^{2}
31 1+(3.516.09i)T+(15.526.8i)T2 1 + (3.51 - 6.09i)T + (-15.5 - 26.8i)T^{2}
37 1+1.40T+37T2 1 + 1.40T + 37T^{2}
41 1+(5.37+9.31i)T+(20.535.5i)T2 1 + (-5.37 + 9.31i)T + (-20.5 - 35.5i)T^{2}
43 1+(5.67+9.83i)T+(21.5+37.2i)T2 1 + (5.67 + 9.83i)T + (-21.5 + 37.2i)T^{2}
47 1+(3.766.52i)T+(23.5+40.7i)T2 1 + (-3.76 - 6.52i)T + (-23.5 + 40.7i)T^{2}
53 111.6T+53T2 1 - 11.6T + 53T^{2}
59 1+(2.22+3.85i)T+(29.551.0i)T2 1 + (-2.22 + 3.85i)T + (-29.5 - 51.0i)T^{2}
61 1+(6.17+10.6i)T+(30.5+52.8i)T2 1 + (6.17 + 10.6i)T + (-30.5 + 52.8i)T^{2}
67 1+(6.33+10.9i)T+(33.558.0i)T2 1 + (-6.33 + 10.9i)T + (-33.5 - 58.0i)T^{2}
71 1+4.93T+71T2 1 + 4.93T + 71T^{2}
73 18.71T+73T2 1 - 8.71T + 73T^{2}
79 1+(0.2800.485i)T+(39.5+68.4i)T2 1 + (-0.280 - 0.485i)T + (-39.5 + 68.4i)T^{2}
83 1+(3.68+6.38i)T+(41.5+71.8i)T2 1 + (3.68 + 6.38i)T + (-41.5 + 71.8i)T^{2}
89 112.1T+89T2 1 - 12.1T + 89T^{2}
97 1+(6.98+12.0i)T+(48.5+84.0i)T2 1 + (6.98 + 12.0i)T + (-48.5 + 84.0i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.217237232050430273013994259126, −8.167135176280835190392007942556, −7.33740177228996824869941513501, −6.79261291213393413620606959807, −5.98722551804189655471573395658, −5.18907297628816534698443414353, −3.97686316805466555623818608643, −3.39991089046578261480340387825, −2.03741848412178840070040362449, −0.19049812709674808531589148802, 0.923916308569710567632737425637, 2.25779341363306444709129470639, 4.16673041092370958248955059272, 4.39830649077907980905875010050, 5.32698404145192458756348121595, 6.12985274750084570377972304229, 7.08650178993723346660949370236, 7.970625843578736775795026261699, 8.484846483616858177209772941595, 9.537280136241147625368058798974

Graph of the ZZ-function along the critical line