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Properties

Label 2-1216-19.11-c1-0-33
Degree $2$
Conductor $1216$
Sign $-0.813 + 0.582i$
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  + (1.5 − 2.59i)3-s + (1 − 1.73i)5-s − 2·7-s + (−3 − 5.19i)9-s + 3·11-s + (1 + 1.73i)13-s + (−3 − 5.19i)15-s + (3 − 5.19i)17-s + (−3.5 − 2.59i)19-s + (−3 + 5.19i)21-s + (2 + 3.46i)23-s + (0.500 + 0.866i)25-s − 9·27-s + (−3 − 5.19i)29-s − 8·31-s + ⋯
L(s)  = 1  + (0.866 − 1.49i)3-s + (0.447 − 0.774i)5-s − 0.755·7-s + (−1 − 1.73i)9-s + 0.904·11-s + (0.277 + 0.480i)13-s + (−0.774 − 1.34i)15-s + (0.727 − 1.26i)17-s + (−0.802 − 0.596i)19-s + (−0.654 + 1.13i)21-s + (0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s − 1.73·27-s + (−0.557 − 0.964i)29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.813 + 0.582i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.154082457\)
\(L(\frac12)\) \(\approx\) \(2.154082457\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.5 + 2.59i)T \)
good3 \( 1 + (-1.5 + 2.59i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5 + 8.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7 - 12.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1 + 1.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.5 + 2.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.252287473325924229936402473775, −8.704909694898883205662825513555, −7.67501456563732144815196020146, −6.98010381012137288010553702983, −6.32503752002096581382086677644, −5.36545148723838319832003762488, −3.93056353121384108972772128180, −2.89369548827754809882945450080, −1.83239589263981842342506649680, −0.840316365765853048516606605503, 2.06183208017385057947220202493, 3.41075162547430933125434319580, 3.54496741903122418953904651114, 4.73229448925364767081950572092, 5.93937603347040367790963498606, 6.54428525324287580171982652725, 7.85957994609888695610652195154, 8.670621924200620095138294850563, 9.380223639214603638906155702403, 10.00431386422429153414374754127

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