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Properties

Label 2-1216-152.125-c1-0-14
Degree $2$
Conductor $1216$
Sign $0.533 - 0.845i$
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  + (0.495 + 0.285i)3-s + (3.31 + 1.91i)5-s − 2.37·7-s + (−1.33 − 2.31i)9-s + 2.62i·11-s + (1.36 − 0.788i)13-s + (1.09 + 1.89i)15-s + (1.53 − 2.65i)17-s + (0.0442 + 4.35i)19-s + (−1.17 − 0.680i)21-s + (4.47 + 7.75i)23-s + (4.82 + 8.35i)25-s − 3.24i·27-s + (8.05 − 4.64i)29-s + 4.38·31-s + ⋯
L(s)  = 1  + (0.285 + 0.165i)3-s + (1.48 + 0.855i)5-s − 0.899·7-s + (−0.445 − 0.771i)9-s + 0.790i·11-s + (0.378 − 0.218i)13-s + (0.282 + 0.489i)15-s + (0.372 − 0.644i)17-s + (0.0101 + 0.999i)19-s + (−0.257 − 0.148i)21-s + (0.933 + 1.61i)23-s + (0.964 + 1.67i)25-s − 0.624i·27-s + (1.49 − 0.863i)29-s + 0.787·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 - 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.533 - 0.845i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.533 - 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.121319182\)
\(L(\frac12)\) \(\approx\) \(2.121319182\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0442 - 4.35i)T \)
good3 \( 1 + (-0.495 - 0.285i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-3.31 - 1.91i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.37T + 7T^{2} \)
11 \( 1 - 2.62iT - 11T^{2} \)
13 \( 1 + (-1.36 + 0.788i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.53 + 2.65i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.47 - 7.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.05 + 4.64i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.38T + 31T^{2} \)
37 \( 1 - 7.00iT - 37T^{2} \)
41 \( 1 + (6.22 - 10.7i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.35 - 1.36i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.15 + 2.00i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.38 + 0.801i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.4 + 6.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.477 + 0.275i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.64 - 4.41i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.63 + 6.29i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.80 - 3.13i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.99 + 3.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.25iT - 83T^{2} \)
89 \( 1 + (4.83 + 8.36i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.67 + 11.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.816631908951293168814032335578, −9.465816373367271671850318808279, −8.344150726832495794104373909767, −7.19951111077461203536731806459, −6.32373789314810465530453853463, −6.01460767923576184614669656574, −4.82145396704180180865597204193, −3.22157675156613083752499927833, −2.95096498941722934946596245587, −1.48825039719016951260939662223, 0.943573904674915825242652430224, 2.29943544564561314099348883642, 3.09385257147955391074142360170, 4.63055006773923007936854883562, 5.44617166920320065922269107254, 6.21315610061924400354253436929, 6.89890624967090099519841763049, 8.388331887128335989419617650039, 8.772617067149584358932715879052, 9.399379519011967428031053968864

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