Properties

Label 2-1216-152.75-c1-0-2
Degree $2$
Conductor $1216$
Sign $-0.783 + 0.621i$
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  + 2i·3-s + 1.73i·5-s + i·7-s − 9-s − 5.19·11-s − 4·13-s − 3.46·15-s + 3·17-s + (−1.73 − 4i)19-s − 2·21-s + 2.00·25-s + 4i·27-s − 6·29-s − 10.3·31-s − 10.3i·33-s + ⋯
L(s)  = 1  + 1.15i·3-s + 0.774i·5-s + 0.377i·7-s − 0.333·9-s − 1.56·11-s − 1.10·13-s − 0.894·15-s + 0.727·17-s + (−0.397 − 0.917i)19-s − 0.436·21-s + 0.400·25-s + 0.769i·27-s − 1.11·29-s − 1.86·31-s − 1.80i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 + 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.783 + 0.621i$
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: \((2,\ 1216,\ (\ :1/2),\ -0.783 + 0.621i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5991258225\)
\(L(\frac12)\) \(\approx\) \(0.5991258225\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 4i)T \)
good3 \( 1 - 2iT - 3T^{2} \)
5 \( 1 - 1.73iT - 5T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 5.19T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 10.3T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 - 5.19T + 43T^{2} \)
47 \( 1 - 9iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 1.73iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 6.92T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + 3.46T + 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 - 3.46iT - 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−10.38239981966888278002393666800, −9.495620166574485751670727448827, −8.882611297471978844250396109963, −7.55294066412374037251147403117, −7.21519852586988781229242746217, −5.70508764640493290913318046038, −5.17859800304580118990284867436, −4.22668199946644908781172766466, −3.09559554929016288658165323050, −2.37288885591025245132387812213, 0.24004515818610407882153837512, 1.60550461373467879002857087036, 2.59392046250544133744023485433, 4.01140617960797528362668159308, 5.19521999778825856942435381932, 5.73152534511079310640977110835, 7.07471507580215251000913315380, 7.58160233552591633159572875816, 8.119483104738470196321745002026, 9.156039137573895883793395697074

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