Properties

Label 2-1216-152.107-c1-0-13
Degree $2$
Conductor $1216$
Sign $0.251 - 0.967i$
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  + (0.132 − 0.0766i)3-s + (1.14 − 0.658i)5-s + 1.40i·7-s + (−1.48 + 2.57i)9-s + 3.27·11-s + (−2.31 + 4.01i)13-s + (0.100 − 0.174i)15-s + (−2.24 − 3.89i)17-s + (2.56 + 3.52i)19-s + (0.107 + 0.186i)21-s + (0.100 + 0.0582i)23-s + (−1.63 + 2.82i)25-s + 0.915i·27-s + (−3.34 + 5.78i)29-s + 2.43·31-s + ⋯
L(s)  = 1  + (0.0766 − 0.0442i)3-s + (0.509 − 0.294i)5-s + 0.531i·7-s + (−0.496 + 0.859i)9-s + 0.987·11-s + (−0.642 + 1.11i)13-s + (0.0260 − 0.0451i)15-s + (−0.545 − 0.945i)17-s + (0.588 + 0.808i)19-s + (0.0235 + 0.0407i)21-s + (0.0210 + 0.0121i)23-s + (−0.326 + 0.565i)25-s + 0.176i·27-s + (−0.620 + 1.07i)29-s + 0.437·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.251 - 0.967i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.251 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.579618384\)
\(L(\frac12)\) \(\approx\) \(1.579618384\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.56 - 3.52i)T \)
good3 \( 1 + (-0.132 + 0.0766i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.14 + 0.658i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.40iT - 7T^{2} \)
11 \( 1 - 3.27T + 11T^{2} \)
13 \( 1 + (2.31 - 4.01i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.24 + 3.89i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.100 - 0.0582i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.34 - 5.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.43T + 31T^{2} \)
37 \( 1 + 8.87T + 37T^{2} \)
41 \( 1 + (-5.96 + 3.44i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.01 - 6.95i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.69 + 4.44i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.47 - 4.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.19 + 5.30i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.55 - 5.51i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.900 - 0.520i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.84 + 4.93i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.104 - 0.181i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.20 - 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 + (5.18 + 2.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.71 - 3.30i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.548810563870828860158557806808, −9.256747653360887794770517434911, −8.434449450836502374179725455676, −7.36639593914860120885971280394, −6.64228272609508852440019691935, −5.53290396673259178359484583666, −4.98279419388309752196870290848, −3.79870210011844501406165356661, −2.48264214429006509080691881866, −1.60617561577685533122359349594, 0.66816546031832207293597835724, 2.24780658688470893280665502259, 3.37519637860659201327757569069, 4.21908292452344205049510752808, 5.46335049658591023259092332433, 6.28319165005619912484162650470, 6.94669652978495273354272717692, 7.956466878193406206774016769070, 8.842403137085033582122745335130, 9.626950972907790836517383085231

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