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Properties

Label 2-936-24.11-c1-0-44
Degree $2$
Conductor $936$
Sign $-0.445 + 0.895i$
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  + (1.18 − 0.768i)2-s + (0.817 − 1.82i)4-s + 0.327·5-s − 2.84i·7-s + (−0.432 − 2.79i)8-s + (0.388 − 0.251i)10-s − 1.48i·11-s + i·13-s + (−2.18 − 3.37i)14-s + (−2.66 − 2.98i)16-s + 0.756i·17-s + 4.98·19-s + (0.267 − 0.597i)20-s + (−1.14 − 1.76i)22-s − 2.71·23-s + ⋯
L(s)  = 1  + (0.839 − 0.543i)2-s + (0.408 − 0.912i)4-s + 0.146·5-s − 1.07i·7-s + (−0.152 − 0.988i)8-s + (0.122 − 0.0795i)10-s − 0.448i·11-s + 0.277i·13-s + (−0.584 − 0.902i)14-s + (−0.665 − 0.746i)16-s + 0.183i·17-s + 1.14·19-s + (0.0598 − 0.133i)20-s + (−0.243 − 0.376i)22-s − 0.566·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.445 + 0.895i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.445 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30636 - 2.10975i\)
\(L(\frac12)\) \(\approx\) \(1.30636 - 2.10975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.18 + 0.768i)T \)
3 \( 1 \)
13 \( 1 - iT \)
good5 \( 1 - 0.327T + 5T^{2} \)
7 \( 1 + 2.84iT - 7T^{2} \)
11 \( 1 + 1.48iT - 11T^{2} \)
17 \( 1 - 0.756iT - 17T^{2} \)
19 \( 1 - 4.98T + 19T^{2} \)
23 \( 1 + 2.71T + 23T^{2} \)
29 \( 1 + 2.16T + 29T^{2} \)
31 \( 1 + 3.62iT - 31T^{2} \)
37 \( 1 + 10.0iT - 37T^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 - 2.76T + 43T^{2} \)
47 \( 1 - 9.45T + 47T^{2} \)
53 \( 1 - 3.68T + 53T^{2} \)
59 \( 1 - 2.06iT - 59T^{2} \)
61 \( 1 + 8.28iT - 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + 1.67T + 71T^{2} \)
73 \( 1 - 4.79T + 73T^{2} \)
79 \( 1 + 1.01iT - 79T^{2} \)
83 \( 1 - 12.2iT - 83T^{2} \)
89 \( 1 - 3.19iT - 89T^{2} \)
97 \( 1 + 5.62T + 97T^{2} \)
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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

−9.891032468254433499908681697805, −9.353703776142647123262275067399, −7.900824304627246998674320135577, −7.14858979635076873508397299783, −6.12872819911073130447096238176, −5.37337422147869390105776073576, −4.17571629330870613920050136569, −3.60981390261697817417635917927, −2.27041003010558306187967881830, −0.904476687823615765022072571351, 2.06392098782560839159957957650, 3.07328321445797813217491973963, 4.18634875056663265017229478840, 5.37812264990684549953961908908, 5.72216315654718349684663485640, 6.86256235216890889237250970294, 7.65165140773834318445543795593, 8.534484530354019293520788664701, 9.348375969274412540941495925969, 10.32186904208729637762256659461

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