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Properties

Label 2-2646-63.20-c1-0-22
Degree 22
Conductor 26462646
Sign 0.870−0.491i0.870 - 0.491i
Analytic cond. 21.128421.1284
Root an. cond. 4.596564.59656
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.0338 + 0.0585i)5-s + 0.999i·8-s + 0.0676i·10-s + (−3.40 − 1.96i)11-s + (3.32 − 1.92i)13-s + (−0.5 + 0.866i)16-s + 1.55·17-s + 5.84i·19-s + (−0.0338 + 0.0585i)20-s + (−1.96 − 3.40i)22-s + (4.78 − 2.76i)23-s + (2.49 − 4.32i)25-s + 3.84·26-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.0151 + 0.0261i)5-s + 0.353i·8-s + 0.0213i·10-s + (−1.02 − 0.592i)11-s + (0.922 − 0.532i)13-s + (−0.125 + 0.216i)16-s + 0.376·17-s + 1.34i·19-s + (−0.00755 + 0.0130i)20-s + (−0.418 − 0.725i)22-s + (0.998 − 0.576i)23-s + (0.499 − 0.865i)25-s + 0.753·26-s + ⋯

Functional equation

Λ(s)=(2646s/2 ΓC(s) L(s)=((0.870−0.491i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2646s/2 ΓC(s+1/2) L(s)=((0.870−0.491i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.870 - 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26462646    =    2⋅33⋅722 \cdot 3^{3} \cdot 7^{2}
Sign: 0.870−0.491i0.870 - 0.491i
Analytic conductor: 21.128421.1284
Root analytic conductor: 4.596564.59656
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2646(1763,⋅)\chi_{2646} (1763, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2646, ( :1/2), 0.870−0.491i)(2,\ 2646,\ (\ :1/2),\ 0.870 - 0.491i)

Particular Values

L(1)L(1) ≈\approx 2.6845906942.684590694
L(12)L(\frac12) ≈\approx 2.6845906942.684590694
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
3 1 1
7 1 1
good5 1+(−0.0338−0.0585i)T+(−2.5+4.33i)T2 1 + (-0.0338 - 0.0585i)T + (-2.5 + 4.33i)T^{2}
11 1+(3.40+1.96i)T+(5.5+9.52i)T2 1 + (3.40 + 1.96i)T + (5.5 + 9.52i)T^{2}
13 1+(−3.32+1.92i)T+(6.5−11.2i)T2 1 + (-3.32 + 1.92i)T + (6.5 - 11.2i)T^{2}
17 1−1.55T+17T2 1 - 1.55T + 17T^{2}
19 1−5.84iT−19T2 1 - 5.84iT - 19T^{2}
23 1+(−4.78+2.76i)T+(11.5−19.9i)T2 1 + (-4.78 + 2.76i)T + (11.5 - 19.9i)T^{2}
29 1+(1.20+0.697i)T+(14.5+25.1i)T2 1 + (1.20 + 0.697i)T + (14.5 + 25.1i)T^{2}
31 1+(−1.09+0.632i)T+(15.5−26.8i)T2 1 + (-1.09 + 0.632i)T + (15.5 - 26.8i)T^{2}
37 1−8.71T+37T2 1 - 8.71T + 37T^{2}
41 1+(−5.17−8.96i)T+(−20.5+35.5i)T2 1 + (-5.17 - 8.96i)T + (-20.5 + 35.5i)T^{2}
43 1+(−0.735+1.27i)T+(−21.5−37.2i)T2 1 + (-0.735 + 1.27i)T + (-21.5 - 37.2i)T^{2}
47 1+(1.77−3.06i)T+(−23.5−40.7i)T2 1 + (1.77 - 3.06i)T + (-23.5 - 40.7i)T^{2}
53 1+7.26iT−53T2 1 + 7.26iT - 53T^{2}
59 1+(−4.70−8.14i)T+(−29.5+51.0i)T2 1 + (-4.70 - 8.14i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.0705−0.0407i)T+(30.5+52.8i)T2 1 + (-0.0705 - 0.0407i)T + (30.5 + 52.8i)T^{2}
67 1+(−7.67−13.2i)T+(−33.5+58.0i)T2 1 + (-7.67 - 13.2i)T + (-33.5 + 58.0i)T^{2}
71 1+4.30iT−71T2 1 + 4.30iT - 71T^{2}
73 1+7.07iT−73T2 1 + 7.07iT - 73T^{2}
79 1+(−3.42+5.92i)T+(−39.5−68.4i)T2 1 + (-3.42 + 5.92i)T + (-39.5 - 68.4i)T^{2}
83 1+(3.93−6.81i)T+(−41.5−71.8i)T2 1 + (3.93 - 6.81i)T + (-41.5 - 71.8i)T^{2}
89 1−11.6T+89T2 1 - 11.6T + 89T^{2}
97 1+(0.363+0.209i)T+(48.5+84.0i)T2 1 + (0.363 + 0.209i)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

−8.568657136159491831677068786271, −8.131941017045725050150658076642, −7.48018817965114455666102849744, −6.34026230352142840873320812017, −5.90393093556884933823605987433, −5.10332735759755685377831061170, −4.19222539618042779795494699969, −3.25666186315684026641677962707, −2.54078302934964964426328638483, −0.965746789316351674721834741205, 0.969110174273724350974627711791, 2.21398704546726729023298272551, 3.06309768176040708452415645919, 3.99323762233507507441169882136, 4.94349760477462411009794594598, 5.43386832154310853629961202877, 6.47954336432207305861835373425, 7.18281376113084403998361035876, 7.916839214754933924352361453620, 9.046608126338897003246847271782

Graph of the ZZ-function along the critical line