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Properties

Label 2-936-104.101-c1-0-21
Degree $2$
Conductor $936$
Sign $0.859 - 0.510i$
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  + (0.203 + 1.39i)2-s + (−1.91 + 0.569i)4-s − 3.13·5-s + (−3.44 − 1.99i)7-s + (−1.18 − 2.56i)8-s + (−0.638 − 4.39i)10-s + (2.10 + 3.64i)11-s + (3.58 − 0.417i)13-s + (2.08 − 5.22i)14-s + (3.35 − 2.18i)16-s + (−0.962 + 1.66i)17-s + (1.67 − 2.89i)19-s + (6.01 − 1.78i)20-s + (−4.67 + 3.69i)22-s + (−1.84 − 3.20i)23-s + ⋯
L(s)  = 1  + (0.143 + 0.989i)2-s + (−0.958 + 0.284i)4-s − 1.40·5-s + (−1.30 − 0.752i)7-s + (−0.419 − 0.907i)8-s + (−0.201 − 1.38i)10-s + (0.635 + 1.10i)11-s + (0.993 − 0.115i)13-s + (0.557 − 1.39i)14-s + (0.837 − 0.545i)16-s + (−0.233 + 0.404i)17-s + (0.383 − 0.664i)19-s + (1.34 − 0.399i)20-s + (−0.997 + 0.787i)22-s + (−0.385 − 0.668i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.859 - 0.510i$
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: \((2,\ 936,\ (\ :1/2),\ 0.859 - 0.510i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833674 + 0.228994i\)
\(L(\frac12)\) \(\approx\) \(0.833674 + 0.228994i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.203 - 1.39i)T \)
3 \( 1 \)
13 \( 1 + (-3.58 + 0.417i)T \)
good5 \( 1 + 3.13T + 5T^{2} \)
7 \( 1 + (3.44 + 1.99i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.10 - 3.64i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.962 - 1.66i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.67 + 2.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.84 + 3.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.70 + 3.29i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.35iT - 31T^{2} \)
37 \( 1 + (0.708 + 1.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.47 + 1.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.0 - 5.81i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.74iT - 47T^{2} \)
53 \( 1 + 2.32iT - 53T^{2} \)
59 \( 1 + (-0.0313 + 0.0543i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.32 - 3.65i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.97 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.28 + 3.05i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 9.02iT - 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + 5.10T + 83T^{2} \)
89 \( 1 + (1.29 - 0.747i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.07 - 2.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.983595045019935471679757958731, −9.092273486437068922485856606115, −8.311847769871414835458255958464, −7.39580643336014545849853315068, −6.82600440800083490992451698167, −6.16094932471851427330204249607, −4.60730735903092048390321575414, −4.01733494498610215454171076719, −3.26502973247979043838250893055, −0.58788125408983383679457654074, 0.876916524486968768873852235279, 2.80922048415789064579953557621, 3.57537910967119300096855136312, 4.14589454192923506854329747787, 5.66488919716536718163714225466, 6.32166453974002038397979721803, 7.68847961057663971627592796732, 8.638157605322136912038332947594, 9.111214430719551170701391808027, 10.03689731746613017115297806945

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