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Properties

Label 2-7644-13.12-c1-0-21
Degree $2$
Conductor $7644$
Sign $-0.979 + 0.200i$
Analytic cond. $61.0376$
Root an. cond. $7.81265$
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  + 3-s + 3.99i·5-s + 9-s + 5.96i·11-s + (3.53 − 0.723i)13-s + 3.99i·15-s − 2.84·17-s + 8.14i·19-s − 5.21·23-s − 10.9·25-s + 27-s − 6.35·29-s − 2.72i·31-s + 5.96i·33-s + 1.80i·37-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.78i·5-s + 0.333·9-s + 1.79i·11-s + (0.979 − 0.200i)13-s + 1.03i·15-s − 0.691·17-s + 1.86i·19-s − 1.08·23-s − 2.19·25-s + 0.192·27-s − 1.18·29-s − 0.489i·31-s + 1.03i·33-s + 0.296i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7644 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7644\)    =    \(2^{2} \cdot 3 \cdot 7^{2} \cdot 13\)
Sign: $-0.979 + 0.200i$
Analytic conductor: \(61.0376\)
Root analytic conductor: \(7.81265\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7644} (4705, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7644,\ (\ :1/2),\ -0.979 + 0.200i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.732656416\)
\(L(\frac12)\) \(\approx\) \(1.732656416\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
7 \( 1 \)
13 \( 1 + (-3.53 + 0.723i)T \)
good5 \( 1 - 3.99iT - 5T^{2} \)
11 \( 1 - 5.96iT - 11T^{2} \)
17 \( 1 + 2.84T + 17T^{2} \)
19 \( 1 - 8.14iT - 19T^{2} \)
23 \( 1 + 5.21T + 23T^{2} \)
29 \( 1 + 6.35T + 29T^{2} \)
31 \( 1 + 2.72iT - 31T^{2} \)
37 \( 1 - 1.80iT - 37T^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 - 1.84T + 43T^{2} \)
47 \( 1 - 6.88iT - 47T^{2} \)
53 \( 1 - 8.83T + 53T^{2} \)
59 \( 1 - 2.88iT - 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 - 0.900iT - 67T^{2} \)
71 \( 1 + 2.56iT - 71T^{2} \)
73 \( 1 + 13.5iT - 73T^{2} \)
79 \( 1 - 2.61T + 79T^{2} \)
83 \( 1 - 0.740iT - 83T^{2} \)
89 \( 1 + 0.921iT - 89T^{2} \)
97 \( 1 + 5.09iT - 97T^{2} \)
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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

−7.921677123678627104833046772127, −7.55481628600529322550243501576, −7.01519724450789640097904378721, −6.18161252399010746239191344252, −5.77456929948337232541310887823, −4.36429372596168361690922664329, −3.84220227740370932242113748422, −3.21777157341194907809390479746, −2.05437133565137635844475990567, −1.92068487911993956843590453065, 0.37361568386646107184105382672, 1.13777073913877521264566573014, 2.11620387265160854158329001630, 3.19673840235667087599283157071, 3.97345206195317985363477641741, 4.57761193757576764692737552318, 5.41057119981333453007226689696, 5.97696821719902353451291716665, 6.77786168562110302122747731509, 7.82366225356477833037422124600

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