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Properties

Label 2-936-24.11-c1-0-6
Degree $2$
Conductor $936$
Sign $0.974 - 0.225i$
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  + (0.339 − 1.37i)2-s + (−1.76 − 0.932i)4-s − 3.97·5-s + 0.281i·7-s + (−1.88 + 2.11i)8-s + (−1.35 + 5.45i)10-s − 4.15i·11-s + i·13-s + (0.386 + 0.0956i)14-s + (2.25 + 3.30i)16-s + 5.36i·17-s + 1.44·19-s + (7.03 + 3.70i)20-s + (−5.70 − 1.41i)22-s − 0.283·23-s + ⋯
L(s)  = 1  + (0.240 − 0.970i)2-s + (−0.884 − 0.466i)4-s − 1.77·5-s + 0.106i·7-s + (−0.665 + 0.746i)8-s + (−0.426 + 1.72i)10-s − 1.25i·11-s + 0.277i·13-s + (0.103 + 0.0255i)14-s + (0.564 + 0.825i)16-s + 1.30i·17-s + 0.331·19-s + (1.57 + 0.828i)20-s + (−1.21 − 0.301i)22-s − 0.0590·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.974 - 0.225i$
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: \((2,\ 936,\ (\ :1/2),\ 0.974 - 0.225i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689495 + 0.0787618i\)
\(L(\frac12)\) \(\approx\) \(0.689495 + 0.0787618i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.339 + 1.37i)T \)
3 \( 1 \)
13 \( 1 - iT \)
good5 \( 1 + 3.97T + 5T^{2} \)
7 \( 1 - 0.281iT - 7T^{2} \)
11 \( 1 + 4.15iT - 11T^{2} \)
17 \( 1 - 5.36iT - 17T^{2} \)
19 \( 1 - 1.44T + 19T^{2} \)
23 \( 1 + 0.283T + 23T^{2} \)
29 \( 1 - 0.381T + 29T^{2} \)
31 \( 1 - 6.98iT - 31T^{2} \)
37 \( 1 - 2.82iT - 37T^{2} \)
41 \( 1 - 5.37iT - 41T^{2} \)
43 \( 1 - 9.55T + 43T^{2} \)
47 \( 1 + 6.24T + 47T^{2} \)
53 \( 1 - 0.416T + 53T^{2} \)
59 \( 1 + 2.68iT - 59T^{2} \)
61 \( 1 + 1.06iT - 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 12.0T + 73T^{2} \)
79 \( 1 - 11.7iT - 79T^{2} \)
83 \( 1 - 1.24iT - 83T^{2} \)
89 \( 1 - 6.64iT - 89T^{2} \)
97 \( 1 + 14.4T + 97T^{2} \)
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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

−10.49964940574297513096843041512, −9.194131020433329988855232899469, −8.406704034634235741800681857736, −7.958504113198086961177001096225, −6.63205627044413830879077098346, −5.51766753058805005637065346831, −4.39329694670563376352439387361, −3.69323035275587815006288268519, −2.93746887239933048618008125573, −1.11109985226902658686879247631, 0.37484924540715085670294070227, 2.93889705636349996082340933183, 4.10240878262658954583541973413, 4.56627897388001102745629939090, 5.63893204929852719408368341528, 7.08614056253004702900689180040, 7.33732719528624835079864039862, 8.025789775844602311653688885630, 8.998404586422866419767897977664, 9.781426153680666442657649105343

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