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Properties

Label 2-1216-19.18-c2-0-15
Degree $2$
Conductor $1216$
Sign $-0.777 - 0.628i$
Analytic cond. $33.1336$
Root an. cond. $5.75617$
Motivic weight $2$
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 − 6.29·5-s + 1.03·7-s + 3.80·9-s − 4.67·11-s − 13.2i·13-s − 14.3i·15-s + 9.09·17-s + (14.7 + 11.9i)19-s + 2.36i·21-s + 16.2·23-s + 14.5·25-s + 29.1i·27-s + 12.4i·29-s + 31.6i·31-s + ⋯
L(s)  = 1  + 0.759i·3-s − 1.25·5-s + 0.148·7-s + 0.423·9-s − 0.424·11-s − 1.01i·13-s − 0.955i·15-s + 0.535·17-s + (0.777 + 0.628i)19-s + 0.112i·21-s + 0.708·23-s + 0.582·25-s + 1.08i·27-s + 0.428i·29-s + 1.02i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.777 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.777 - 0.628i$
Analytic conductor: \(33.1336\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1),\ -0.777 - 0.628i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9678084512\)
\(L(\frac12)\) \(\approx\) \(0.9678084512\)
\(L(2)\) 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 \( 1 + (-14.7 - 11.9i)T \)
good3 \( 1 - 2.27iT - 9T^{2} \)
5 \( 1 + 6.29T + 25T^{2} \)
7 \( 1 - 1.03T + 49T^{2} \)
11 \( 1 + 4.67T + 121T^{2} \)
13 \( 1 + 13.2iT - 169T^{2} \)
17 \( 1 - 9.09T + 289T^{2} \)
23 \( 1 - 16.2T + 529T^{2} \)
29 \( 1 - 12.4iT - 841T^{2} \)
31 \( 1 - 31.6iT - 961T^{2} \)
37 \( 1 - 57.9iT - 1.36e3T^{2} \)
41 \( 1 + 33.8iT - 1.68e3T^{2} \)
43 \( 1 + 31.3T + 1.84e3T^{2} \)
47 \( 1 + 14.7T + 2.20e3T^{2} \)
53 \( 1 + 2.57iT - 2.80e3T^{2} \)
59 \( 1 + 43.3iT - 3.48e3T^{2} \)
61 \( 1 + 50.5T + 3.72e3T^{2} \)
67 \( 1 + 83.1iT - 4.48e3T^{2} \)
71 \( 1 + 8.54iT - 5.04e3T^{2} \)
73 \( 1 + 145.T + 5.32e3T^{2} \)
79 \( 1 - 100. iT - 6.24e3T^{2} \)
83 \( 1 - 139.T + 6.88e3T^{2} \)
89 \( 1 - 109. iT - 7.92e3T^{2} \)
97 \( 1 - 81.1iT - 9.40e3T^{2} \)
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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

−10.05136839341474151540081629173, −9.058247113617055243306342619607, −8.049236273675236703287975969435, −7.67106578121957633759452927839, −6.67450197942295894963328821815, −5.25953092497327948266776928881, −4.80750695806923664009953209613, −3.59923014515697373346394729304, −3.17741767857059110370288498631, −1.21154715292055869897683635847, 0.32674498055136161927044966236, 1.57554404245006103713891167580, 2.89418720729363149261037607101, 4.03898177724056554068049221001, 4.75827628408027894824969853671, 5.99761751074204186741019747130, 7.09053246927243930883849395508, 7.46726121599538630703877709613, 8.153141806302645485776870688263, 9.134762244833007404777146388569

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