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Properties

Label 2-1216-76.7-c0-0-0
Degree $2$
Conductor $1216$
Sign $0.305 + 0.952i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)13-s + (0.866 + 0.499i)15-s + (−0.5 − 0.866i)17-s + i·19-s + (−0.866 − 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − 2i·31-s + 0.999i·39-s + (−0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + (0.866 + 0.5i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.5 − 0.866i)5-s + (0.5 − 0.866i)13-s + (0.866 + 0.499i)15-s + (−0.5 − 0.866i)17-s + i·19-s + (−0.866 − 0.5i)23-s − i·27-s + (0.5 − 0.866i)29-s − 2i·31-s + 0.999i·39-s + (−0.5 − 0.866i)41-s + (0.866 − 0.5i)43-s + (0.866 + 0.5i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.305 + 0.952i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :0),\ 0.305 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5788813692\)
\(L(\frac12)\) \(\approx\) \(0.5788813692\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + 2iT - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.968958838414498716776928380114, −8.905033504640740222363836905380, −8.165910218485287290731966057280, −7.46601906877010043582567352113, −6.00095206692150054700176195975, −5.66622558563739064601229591529, −4.51414379946950718010219316691, −4.05246587633755526302103191773, −2.46690302912986844679491042576, −0.59137151464952032825805018407, 1.50863776185989750114429940837, 3.01058750954397814600227403540, 3.98271308439624285514057492539, 5.10737313499839080746870562739, 6.18390062174837916117022491537, 6.75182038799543721294901214089, 7.29342028013455314973685785391, 8.496181170385586814556598231387, 9.168610146115285180020975984204, 10.45468339489145265525669036034

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