Properties

Label 2-1216-76.75-c1-0-30
Degree $2$
Conductor $1216$
Sign $-0.844 + 0.535i$
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  − 2.35·3-s + 2.56·5-s − 4.15i·7-s + 2.56·9-s − 2.33i·11-s − 4.29i·13-s − 6.04·15-s − 17-s + (−3.68 + 2.33i)19-s + 9.79i·21-s + 1.82i·23-s + 1.56·25-s + 1.03·27-s − 1.20i·29-s − 1.32·31-s + ⋯
L(s)  = 1  − 1.36·3-s + 1.14·5-s − 1.56i·7-s + 0.853·9-s − 0.703i·11-s − 1.19i·13-s − 1.55·15-s − 0.242·17-s + (−0.844 + 0.535i)19-s + 2.13i·21-s + 0.379i·23-s + 0.312·25-s + 0.198·27-s − 0.223i·29-s − 0.237·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.844 + 0.535i$
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: \((2,\ 1216,\ (\ :1/2),\ -0.844 + 0.535i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7470193398\)
\(L(\frac12)\) \(\approx\) \(0.7470193398\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.68 - 2.33i)T \)
good3 \( 1 + 2.35T + 3T^{2} \)
5 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 + 4.15iT - 7T^{2} \)
11 \( 1 + 2.33iT - 11T^{2} \)
13 \( 1 + 4.29iT - 13T^{2} \)
17 \( 1 + T + 17T^{2} \)
23 \( 1 - 1.82iT - 23T^{2} \)
29 \( 1 + 1.20iT - 29T^{2} \)
31 \( 1 + 1.32T + 31T^{2} \)
37 \( 1 - 5.49iT - 37T^{2} \)
41 \( 1 - 5.49iT - 41T^{2} \)
43 \( 1 - 1.30iT - 43T^{2} \)
47 \( 1 + 6.99iT - 47T^{2} \)
53 \( 1 + 9.79iT - 53T^{2} \)
59 \( 1 + 6.33T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 0.290T + 67T^{2} \)
71 \( 1 - 2.06T + 71T^{2} \)
73 \( 1 + 0.123T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 + 14.0iT - 89T^{2} \)
97 \( 1 + 2.41iT - 97T^{2} \)
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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

−9.930669405528168404247684300554, −8.538201240945928620214282917165, −7.60791569358433734416185114470, −6.59455593105475762006348282665, −6.05010660373898966844526961538, −5.33044582128154813133030455185, −4.40602227316094650033305670253, −3.21718395778141571250310120246, −1.51556262701591832727803067715, −0.37599194479412328489182952397, 1.77250422970883646264766990988, 2.50888245606522198657795128554, 4.44575738756450146125143018283, 5.14079191344998227406838762423, 6.07165129904256364124301589247, 6.25805383625151942455209160977, 7.31217727263245740332433123279, 8.968707626827045052702916154550, 9.074063374172853770410788903704, 10.13346505932965829194924797960

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