Properties

Label 2-1216-76.75-c1-0-1
Degree $2$
Conductor $1216$
Sign $-i$
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  − 4.27·5-s − 4.77i·7-s − 3·9-s − 2.15i·11-s + 0.274·17-s + 4.35i·19-s + 8.71i·23-s + 13.2·25-s + 20.4i·35-s − 7.40i·43-s + 12.8·45-s + 9.07i·47-s − 15.8·49-s + 9.19i·55-s + 3.72·61-s + ⋯
L(s)  = 1  − 1.91·5-s − 1.80i·7-s − 9-s − 0.648i·11-s + 0.0666·17-s + 0.999i·19-s + 1.81i·23-s + 2.65·25-s + 3.45i·35-s − 1.12i·43-s + 1.91·45-s + 1.32i·47-s − 2.26·49-s + 1.23i·55-s + 0.476·61-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3151947294\)
\(L(\frac12)\) \(\approx\) \(0.3151947294\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.35iT \)
good3 \( 1 + 3T^{2} \)
5 \( 1 + 4.27T + 5T^{2} \)
7 \( 1 + 4.77iT - 7T^{2} \)
11 \( 1 + 2.15iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 0.274T + 17T^{2} \)
23 \( 1 - 8.71iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 7.40iT - 43T^{2} \)
47 \( 1 - 9.07iT - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 3.72T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16.8T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 8.71iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.13032840776092542420332590181, −8.916929548263368977706596870749, −8.021907053161617209527506853681, −7.64781684464786081910432226637, −6.93169628705816716995807753944, −5.71527688299552490967522665542, −4.52393997958540853495981910957, −3.66776447146328898094091819872, −3.31322421222792031243241309871, −1.01095105835856636341406806106, 0.16788113180149400012379331689, 2.46721256464687153039851315276, 3.14026589385116810095749408704, 4.43211591071525013938648238598, 5.11068542833249976671561847951, 6.22874240249406335119383773593, 7.11976562377767559781329086965, 8.153796937980897713385019665524, 8.599305754364492006218002211911, 9.160486899394893611915655681478

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