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Properties

Label 2-936-104.19-c1-0-25
Degree $2$
Conductor $936$
Sign $0.999 + 0.0193i$
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  + (−1.36 − 0.366i)2-s + (1.73 + i)4-s + (0.366 + 0.366i)5-s + (2.36 + 0.633i)7-s + (−1.99 − 2i)8-s + (−0.366 − 0.633i)10-s + (3.36 − 0.901i)11-s + (3.5 + 0.866i)13-s + (−3 − 1.73i)14-s + (1.99 + 3.46i)16-s + (−0.232 + 0.133i)17-s + (−4.09 − 1.09i)19-s + (0.267 + i)20-s − 4.92·22-s + (−0.366 + 0.633i)23-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (0.163 + 0.163i)5-s + (0.894 + 0.239i)7-s + (−0.707 − 0.707i)8-s + (−0.115 − 0.200i)10-s + (1.01 − 0.271i)11-s + (0.970 + 0.240i)13-s + (−0.801 − 0.462i)14-s + (0.499 + 0.866i)16-s + (−0.0562 + 0.0324i)17-s + (−0.940 − 0.251i)19-s + (0.0599 + 0.223i)20-s − 1.05·22-s + (−0.0763 + 0.132i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0193i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.999 + 0.0193i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.999 + 0.0193i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28309 - 0.0124324i\)
\(L(\frac12)\) \(\approx\) \(1.28309 - 0.0124324i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
3 \( 1 \)
13 \( 1 + (-3.5 - 0.866i)T \)
good5 \( 1 + (-0.366 - 0.366i)T + 5iT^{2} \)
7 \( 1 + (-2.36 - 0.633i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-3.36 + 0.901i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (0.232 - 0.133i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.09 + 1.09i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.366 - 0.633i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.59 - 1.5i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 - 4i)T + 31iT^{2} \)
37 \( 1 + (1.86 + 6.96i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.5 - 9.33i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.633 - 0.366i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.26 - 2.26i)T - 47iT^{2} \)
53 \( 1 - 8.66iT - 53T^{2} \)
59 \( 1 + (-2.80 + 10.4i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.66 + 0.964i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.56 + 5.83i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.83 - 6.83i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-7.83 - 7.83i)T + 73iT^{2} \)
79 \( 1 + 1.07iT - 79T^{2} \)
83 \( 1 + (-7.92 + 7.92i)T - 83iT^{2} \)
89 \( 1 + (-12.5 + 3.36i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-15.2 - 4.09i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.07517185532528795565022375595, −9.031230584776193848223218845113, −8.565478890778972911063834607184, −7.81896796485936780596292093304, −6.58530209257513790396002699107, −6.17661322863706833767931815444, −4.62774120428496797313902698109, −3.53280716856945501875914399351, −2.22219898721644570085246962603, −1.16561292236569533570320317910, 1.09506479878967122038639681596, 2.07755763484233121765089651466, 3.71447979056261487683136555976, 4.89552009356364768004235383480, 6.02669130431750697348322653062, 6.70984596219345152033879042557, 7.72721833474217809237227381476, 8.474845699075002009081221021657, 9.048695626822718283921603721775, 10.03397687153176263049091506521

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