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Properties

Label 2-1216-76.31-c1-0-13
Degree $2$
Conductor $1216$
Sign $0.974 - 0.222i$
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  + (0.305 + 0.528i)3-s + (−1.59 − 2.75i)5-s + 2.36i·7-s + (1.31 − 2.27i)9-s + 5.46i·11-s + (2.31 + 1.33i)13-s + (0.971 − 1.68i)15-s + (−0.552 − 0.957i)17-s + (−1.37 − 4.13i)19-s + (−1.24 + 0.720i)21-s + (2.46 + 1.42i)23-s + (−2.57 + 4.46i)25-s + 3.43·27-s + (5.63 + 3.25i)29-s + 1.01·31-s + ⋯
L(s)  = 1  + (0.176 + 0.305i)3-s + (−0.712 − 1.23i)5-s + 0.893i·7-s + (0.437 − 0.758i)9-s + 1.64i·11-s + (0.643 + 0.371i)13-s + (0.250 − 0.434i)15-s + (−0.134 − 0.232i)17-s + (−0.316 − 0.948i)19-s + (−0.272 + 0.157i)21-s + (0.513 + 0.296i)23-s + (−0.514 + 0.892i)25-s + 0.660·27-s + (1.04 + 0.604i)29-s + 0.182·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.974 - 0.222i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.974 - 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.613227637\)
\(L(\frac12)\) \(\approx\) \(1.613227637\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.37 + 4.13i)T \)
good3 \( 1 + (-0.305 - 0.528i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.59 + 2.75i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 2.36iT - 7T^{2} \)
11 \( 1 - 5.46iT - 11T^{2} \)
13 \( 1 + (-2.31 - 1.33i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.552 + 0.957i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2.46 - 1.42i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.63 - 3.25i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.01T + 31T^{2} \)
37 \( 1 - 0.450iT - 37T^{2} \)
41 \( 1 + (-0.336 + 0.194i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.96 + 2.86i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.91 - 1.68i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.53 - 2.03i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.82 - 11.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.77 + 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.27 + 7.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.07 - 1.86i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.91 - 6.78i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.57 + 9.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.14iT - 83T^{2} \)
89 \( 1 + (-4.19 - 2.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.641 - 0.370i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.406494048585345217755116736007, −9.040383673355449493566471334869, −8.406421134717823629249745723439, −7.28640376174527422467256990870, −6.55772624891666170416023108940, −5.22757972099651206888870742811, −4.57627878100346871250014223896, −3.88352876759695211646979256546, −2.45180527208939842868713003167, −1.06039411880304173487539663637, 0.907202853264175129552706536982, 2.59522486721728578693300738131, 3.52422571431006747144923502725, 4.20588463126734674923052319824, 5.67599942130219062127587902708, 6.56828564491659444012974343453, 7.23385594805841599856045740109, 8.128016120950141266878141323846, 8.447133588618260757867622428812, 10.02312613072430396577888081315

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