Properties

Label 2-1216-152.75-c1-0-28
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  − 1.31i·5-s − 2.27i·7-s + 3·9-s + 2.15·11-s + 0.274·17-s − 4.35·19-s − 4i·23-s + 3.27·25-s − 2.98·35-s − 7.40·43-s − 3.94i·45-s − 10.2i·47-s + 1.82·49-s − 2.82i·55-s − 15.1i·61-s + ⋯
L(s)  = 1  − 0.587i·5-s − 0.859i·7-s + 9-s + 0.648·11-s + 0.0666·17-s − 1.00·19-s − 0.834i·23-s + 0.654·25-s − 0.505·35-s − 1.12·43-s − 0.587i·45-s − 1.49i·47-s + 0.260·49-s − 0.380i·55-s − 1.94i·61-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: $\chi_{1216} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.709066703\)
\(L(\frac12)\) \(\approx\) \(1.709066703\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.35T \)
good3 \( 1 - 3T^{2} \)
5 \( 1 + 1.31iT - 5T^{2} \)
7 \( 1 + 2.27iT - 7T^{2} \)
11 \( 1 - 2.15T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 0.274T + 17T^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 7.40T + 43T^{2} \)
47 \( 1 + 10.2iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 15.1iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 16.8T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 8.71T + 83T^{2} \)
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

−9.610829495689491885157007064433, −8.727775274015526350937035002192, −7.997745379862870448048105140353, −6.92067259333288725101590637426, −6.51509447052524739276851964597, −5.09014521127127939059030055556, −4.35457860316717343780127143069, −3.61235822360394227683251533727, −1.96041085499069111032198366682, −0.793777972022043025629662754543, 1.51094435064327148319910965400, 2.65904015033544259890180611304, 3.77620183315409154081257765831, 4.71582781314895692681393336797, 5.82479435740534593051797641339, 6.63120951833219185302824088883, 7.30251028723848541468153465015, 8.341145618220282151511746010395, 9.141081149691055205125468668619, 9.867232813448841311021774088648

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