Properties

Label 2-1216-152.75-c1-0-3
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  − 3.40i·3-s + 4.62i·7-s − 8.56·9-s − 0.223·13-s − 3.69·17-s + 4.35i·19-s + 15.7·21-s + 6.49i·23-s + 5·25-s + 18.9i·27-s − 6.57·29-s + 8.71·37-s + 0.761i·39-s + 6i·47-s − 14.4·49-s + ⋯
L(s)  = 1  − 1.96i·3-s + 1.74i·7-s − 2.85·9-s − 0.0620·13-s − 0.896·17-s + 0.999i·19-s + 3.43·21-s + 1.35i·23-s + 25-s + 3.64i·27-s − 1.22·29-s + 1.43·37-s + 0.121i·39-s + 0.875i·47-s − 2.06·49-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

\(L(1)\) \(\approx\) \(0.8591246126\)
\(L(\frac12)\) \(\approx\) \(0.8591246126\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.40iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 4.62iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 0.223T + 13T^{2} \)
17 \( 1 + 3.69T + 17T^{2} \)
23 \( 1 - 6.49iT - 23T^{2} \)
29 \( 1 + 6.57T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 8.71T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 + 13.8T + 53T^{2} \)
59 \( 1 - 2.95iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 10.6iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 1.82T + 73T^{2} \)
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

−9.402970185345317384614524803689, −8.898347554196132577145659278944, −8.030179454810842379354939931149, −7.49474376054674785617231408714, −6.35685515594104635607487324737, −5.97081154396229275917817415892, −5.12552490668844645305537566303, −3.19374490205225294880471124287, −2.30676200482846665774037759052, −1.52248119877635404466962452961, 0.35649047348992046243818960061, 2.72759647693396293446416429936, 3.74049200610976929900937415185, 4.47233484491098425370201172495, 4.88553479086233434933661687792, 6.20707242935184470864951439469, 7.12344838332619782226158857662, 8.258107895054660166040459255119, 9.041291154394641287401591917419, 9.758983556537289394013249798144

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