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Properties

Label 2-936-24.11-c1-0-7
Degree $2$
Conductor $936$
Sign $0.702 - 0.711i$
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  + (0.634 − 1.26i)2-s + (−1.19 − 1.60i)4-s − 0.743·5-s + 1.72i·7-s + (−2.78 + 0.494i)8-s + (−0.471 + 0.939i)10-s + 4.66i·11-s − i·13-s + (2.17 + 1.09i)14-s + (−1.14 + 3.83i)16-s + 6.58i·17-s − 3.02·19-s + (0.888 + 1.19i)20-s + (5.90 + 2.96i)22-s − 7.02·23-s + ⋯
L(s)  = 1  + (0.448 − 0.893i)2-s + (−0.597 − 0.801i)4-s − 0.332·5-s + 0.650i·7-s + (−0.984 + 0.174i)8-s + (−0.149 + 0.297i)10-s + 1.40i·11-s − 0.277i·13-s + (0.581 + 0.291i)14-s + (−0.285 + 0.958i)16-s + 1.59i·17-s − 0.694·19-s + (0.198 + 0.266i)20-s + (1.25 + 0.631i)22-s − 1.46·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.702 - 0.711i$
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: \((2,\ 936,\ (\ :1/2),\ 0.702 - 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.994991 + 0.415561i\)
\(L(\frac12)\) \(\approx\) \(0.994991 + 0.415561i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.634 + 1.26i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 + 0.743T + 5T^{2} \)
7 \( 1 - 1.72iT - 7T^{2} \)
11 \( 1 - 4.66iT - 11T^{2} \)
17 \( 1 - 6.58iT - 17T^{2} \)
19 \( 1 + 3.02T + 19T^{2} \)
23 \( 1 + 7.02T + 23T^{2} \)
29 \( 1 - 10.3T + 29T^{2} \)
31 \( 1 + 0.619iT - 31T^{2} \)
37 \( 1 + 0.689iT - 37T^{2} \)
41 \( 1 - 5.53iT - 41T^{2} \)
43 \( 1 - 8.98T + 43T^{2} \)
47 \( 1 - 0.183T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 - 13.2iT - 59T^{2} \)
61 \( 1 + 13.1iT - 61T^{2} \)
67 \( 1 - 0.268T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 2.74T + 73T^{2} \)
79 \( 1 - 11.0iT - 79T^{2} \)
83 \( 1 - 6.15iT - 83T^{2} \)
89 \( 1 + 13.1iT - 89T^{2} \)
97 \( 1 + 17.2T + 97T^{2} \)
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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

−10.21235085352156640259396130510, −9.640002443368506416118959972104, −8.555773946210586209074366919087, −7.88249912192005262735785516493, −6.43796960745317094418863470562, −5.77470074694822477926116667305, −4.52952824116532085170170017389, −3.99604082453962127420146577136, −2.60290115959495994678813236286, −1.70018884539146354692855439991, 0.43106896683224352642958828929, 2.79778377227671459187680903248, 3.86148639588902986067796074320, 4.61153468978210408330739827200, 5.75171741419170744900063496420, 6.48309831118968869160790090600, 7.40115848753425770966757742231, 8.125242195641575689167635425000, 8.848857191839653073622878810609, 9.811436307211602263225496091361

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