Properties

Label 2-1216-8.5-c1-0-35
Degree $2$
Conductor $1216$
Sign $-0.258 - 0.965i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
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  − 3.37i·3-s − 2.52i·5-s − 3.31·7-s − 8.37·9-s + 2.37i·11-s − 5.84i·13-s − 8.51·15-s + 5·17-s + i·19-s + 11.1i·21-s − 0.792·23-s − 1.37·25-s + 18.1i·27-s + 2.67i·29-s − 3.46·31-s + ⋯
L(s)  = 1  − 1.94i·3-s − 1.12i·5-s − 1.25·7-s − 2.79·9-s + 0.715i·11-s − 1.61i·13-s − 2.19·15-s + 1.21·17-s + 0.229i·19-s + 2.44i·21-s − 0.165·23-s − 0.274·25-s + 3.48i·27-s + 0.496i·29-s − 0.622·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.258 - 0.965i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.258 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7560444087\)
\(L(\frac12)\) \(\approx\) \(0.7560444087\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 - iT \)
good3 \( 1 + 3.37iT - 3T^{2} \)
5 \( 1 + 2.52iT - 5T^{2} \)
7 \( 1 + 3.31T + 7T^{2} \)
11 \( 1 - 2.37iT - 11T^{2} \)
13 \( 1 + 5.84iT - 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
23 \( 1 + 0.792T + 23T^{2} \)
29 \( 1 - 2.67iT - 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + 10.0iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 3.62iT - 43T^{2} \)
47 \( 1 - 0.644T + 47T^{2} \)
53 \( 1 + 6.13iT - 53T^{2} \)
59 \( 1 - 1.37iT - 59T^{2} \)
61 \( 1 - 14.5iT - 61T^{2} \)
67 \( 1 - 2.62iT - 67T^{2} \)
71 \( 1 - 6.63T + 71T^{2} \)
73 \( 1 - 8.48T + 73T^{2} \)
79 \( 1 - 0.294T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 + 15.4T + 89T^{2} \)
97 \( 1 + 2.74T + 97T^{2} \)
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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

−8.939797706059125665078396854096, −8.158511881012632733825600450330, −7.50044481868229958723669777156, −6.82395277533586847794729196559, −5.68898952834443705400726762716, −5.43997765691192411642242726781, −3.55694211544024947910049892385, −2.57667518255882584701265560532, −1.29382277824264498954001332178, −0.33758132492753181000324643856, 2.74784087615990220503425551302, 3.38000801168883427943071228047, 4.02578017180304037392457850052, 5.14746427976961150171856336284, 6.17339158475382890706296268520, 6.66845416519706802794804366551, 8.069636940753382475288799253180, 9.097462480631607931074309237382, 9.678536998112418118376831835959, 10.08896169482734412732667088776

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