Properties

Label 2-1216-152.75-c1-0-31
Degree $2$
Conductor $1216$
Sign $-0.707 + 0.707i$
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  − 2.27i·3-s − 4.53i·7-s − 2.16·9-s + 6.13·13-s + 8.23·17-s − 4.35i·19-s − 10.3·21-s + 2.86i·23-s + 5·25-s − 1.90i·27-s − 10.6·29-s − 8.71·37-s − 13.9i·39-s + 6i·47-s − 13.5·49-s + ⋯
L(s)  = 1  − 1.31i·3-s − 1.71i·7-s − 0.721·9-s + 1.70·13-s + 1.99·17-s − 0.999i·19-s − 2.24·21-s + 0.596i·23-s + 25-s − 0.365i·27-s − 1.98·29-s − 1.43·37-s − 2.23i·39-s + 0.875i·47-s − 1.93·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.707 + 0.707i$
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: \((2,\ 1216,\ (\ :1/2),\ -0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.812183733\)
\(L(\frac12)\) \(\approx\) \(1.812183733\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.27iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 4.53iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 6.13T + 13T^{2} \)
17 \( 1 - 8.23T + 17T^{2} \)
23 \( 1 - 2.86iT - 23T^{2} \)
29 \( 1 + 10.6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 8.71T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 2.95T + 53T^{2} \)
59 \( 1 - 14.5iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 5.44iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 15.6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
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

−9.381996715458224787635908422540, −8.342229691596337895063183955525, −7.51786776187290472538472034500, −7.17789144359820282130489966047, −6.30359997117693088734792549716, −5.36551815793862251478044758222, −3.96793445864777943370344335218, −3.24743475384073925113687777647, −1.47906090322538787854137960859, −0.902363341602461692473437675316, 1.73112876918739201559841519292, 3.31312460625448148879951640577, 3.68843535605050506794327344571, 5.20190142582109717149918718320, 5.52627515693596945779892259123, 6.41140979217804061323318487696, 7.920721536851764556932490704807, 8.638629013733774280933063823047, 9.232948300642814999252332688385, 9.959504149547157642915157542780

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