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Properties

Label 2-936-104.101-c1-0-8
Degree $2$
Conductor $936$
Sign $0.0458 - 0.998i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.662 − 1.24i)2-s + (−1.12 + 1.65i)4-s − 1.88·5-s + (3.80 + 2.19i)7-s + (2.81 + 0.308i)8-s + (1.24 + 2.35i)10-s + (0.0662 + 0.114i)11-s + (−3.59 + 0.274i)13-s + (0.226 − 6.21i)14-s + (−1.47 − 3.71i)16-s + (−1.14 + 1.98i)17-s + (−2.18 + 3.79i)19-s + (2.11 − 3.12i)20-s + (0.0995 − 0.158i)22-s + (−1.97 − 3.42i)23-s + ⋯
L(s)  = 1  + (−0.468 − 0.883i)2-s + (−0.561 + 0.827i)4-s − 0.843·5-s + (1.43 + 0.831i)7-s + (0.994 + 0.108i)8-s + (0.394 + 0.745i)10-s + (0.0199 + 0.0346i)11-s + (−0.997 + 0.0760i)13-s + (0.0604 − 1.66i)14-s + (−0.369 − 0.929i)16-s + (−0.278 + 0.482i)17-s + (−0.502 + 0.870i)19-s + (0.473 − 0.697i)20-s + (0.0212 − 0.0338i)22-s + (−0.411 − 0.713i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0458 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0458 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.0458 - 0.998i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.0458 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401672 + 0.383648i\)
\(L(\frac12)\) \(\approx\) \(0.401672 + 0.383648i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.662 + 1.24i)T \)
3 \( 1 \)
13 \( 1 + (3.59 - 0.274i)T \)
good5 \( 1 + 1.88T + 5T^{2} \)
7 \( 1 + (-3.80 - 2.19i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0662 - 0.114i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.14 - 1.98i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.18 - 3.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.97 + 3.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.20 - 1.27i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.06iT - 31T^{2} \)
37 \( 1 + (5.28 + 9.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.14 - 4.12i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.32 - 4.80i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.1iT - 47T^{2} \)
53 \( 1 - 8.28iT - 53T^{2} \)
59 \( 1 + (1.65 - 2.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.40 - 3.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.540 - 0.936i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.45 + 3.72i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.14iT - 73T^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 + 0.144T + 83T^{2} \)
89 \( 1 + (-14.2 + 8.20i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.11 - 3.53i)T + (48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.48169630389081571567097957957, −9.376518719411874153710376410118, −8.570303168995295291263961085360, −7.968153174990848918274353906478, −7.36165294729265886812806364283, −5.75460327619157187799174022703, −4.62713589941771718457595939928, −4.03213724793448229695748037965, −2.58268789772641719087183097077, −1.66610626150892231181042725178, 0.30584582048944979862056504181, 1.88088496953695637926302373052, 3.87095318724732141821396763002, 4.73440850504500656858992977030, 5.33300802266094458658133814262, 6.89084242938221293056844021809, 7.33289402486176620709187555592, 8.090356573147867893760231929905, 8.694306156188709517032273521711, 9.815938846228131620183166979136

Graph of the $Z$-function along the critical line