Properties

Label 2-1216-8.5-c1-0-23
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 − 4.77·7-s + 3·9-s + 2.27i·11-s − 6.09i·13-s − 4.27·17-s i·19-s − 3.46·23-s + 3.27·25-s − 6.09i·29-s + 2.62·31-s − 6.27i·35-s − 0.837i·37-s − 10.5·41-s − 10.2i·43-s + ⋯
L(s)  = 1  + 0.587i·5-s − 1.80·7-s + 9-s + 0.685i·11-s − 1.68i·13-s − 1.03·17-s − 0.229i·19-s − 0.722·23-s + 0.654·25-s − 1.13i·29-s + 0.471·31-s − 1.06i·35-s − 0.137i·37-s − 1.64·41-s − 1.56i·43-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: $\chi_{1216} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7085206655\)
\(L(\frac12)\) \(\approx\) \(0.7085206655\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 - 3T^{2} \)
5 \( 1 - 1.31iT - 5T^{2} \)
7 \( 1 + 4.77T + 7T^{2} \)
11 \( 1 - 2.27iT - 11T^{2} \)
13 \( 1 + 6.09iT - 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 + 6.09iT - 29T^{2} \)
31 \( 1 - 2.62T + 31T^{2} \)
37 \( 1 + 0.837iT - 37T^{2} \)
41 \( 1 + 10.5T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 - 4.77T + 47T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + 8.54iT - 59T^{2} \)
61 \( 1 - 1.31iT - 61T^{2} \)
67 \( 1 + 8.54iT - 67T^{2} \)
71 \( 1 + 14.8T + 71T^{2} \)
73 \( 1 - 4.27T + 73T^{2} \)
79 \( 1 + 4.30T + 79T^{2} \)
83 \( 1 - 13.0iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 1.45T + 97T^{2} \)
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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

−9.873934944705942616920154244955, −8.772385888071292859194367618119, −7.68851552477435339532462936424, −6.82882357071684882543076346199, −6.45176034654991702969123973796, −5.31922458184758662245751619950, −4.09006766000337156255922678264, −3.24542229861721662217447194214, −2.29672218191831991215928374594, −0.30353064188474975068237736203, 1.41934589845176613582732728408, 2.86706302912044422922297189904, 3.96680065718833501099000171312, 4.63224344237081132587438553932, 6.02632925095336370116787610235, 6.64829817154266407175693130257, 7.24416439246320593922193610161, 8.719253731781574709841305119728, 9.075411287014856530232978993717, 9.887924037948746166785394981526

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