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Properties

Label 2-912-19.7-c1-0-14
Degree $2$
Conductor $912$
Sign $-0.321 + 0.946i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (−0.5 − 0.866i)3-s − 7-s + (−0.499 + 0.866i)9-s + 2·11-s + (1.5 − 2.59i)13-s + (−2 − 3.46i)17-s + (−4 + 1.73i)19-s + (0.5 + 0.866i)21-s + (2 − 3.46i)23-s + (2.5 − 4.33i)25-s + 0.999·27-s + 3·31-s + (−1 − 1.73i)33-s − 5·37-s − 3·39-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s − 0.377·7-s + (−0.166 + 0.288i)9-s + 0.603·11-s + (0.416 − 0.720i)13-s + (−0.485 − 0.840i)17-s + (−0.917 + 0.397i)19-s + (0.109 + 0.188i)21-s + (0.417 − 0.722i)23-s + (0.5 − 0.866i)25-s + 0.192·27-s + 0.538·31-s + (−0.174 − 0.301i)33-s − 0.821·37-s − 0.480·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.321 + 0.946i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.321 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.626322 - 0.874553i\)
\(L(\frac12)\) \(\approx\) \(0.626322 - 0.874553i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (2 + 3.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.5 + 7.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7 - 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8T + 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.993617443608869853442800013521, −8.776645008710452617809812922838, −8.329606155249876670375813681076, −7.01332594763960408665411084164, −6.57377424807135374838341145114, −5.57814408467689425737300996369, −4.53488921649565395276253547459, −3.34551188670806943062898976178, −2.13091480285622525382509842280, −0.54194631215494794124172825010, 1.54886931932192174762563354619, 3.14962463111515055093167669483, 4.11601781371529114005569547984, 4.94932860178085626855492071007, 6.25237145859147901907165990483, 6.61165737228390701623702765014, 7.87839890661223125054854045295, 9.021324787538220114908623211405, 9.296795405918880048668634561667, 10.47140252490331208747956144629

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