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Properties

Label 2-912-12.11-c1-0-2
Degree $2$
Conductor $912$
Sign $-0.674 + 0.738i$
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  + (−0.372 + 1.69i)3-s + 4.46i·5-s − 2.16i·7-s + (−2.72 − 1.26i)9-s − 2.80·11-s − 4.91·13-s + (−7.55 − 1.66i)15-s − 2.05i·17-s − i·19-s + (3.66 + 0.806i)21-s + 5.97·23-s − 14.9·25-s + (3.14 − 4.13i)27-s + 7.58i·29-s − 6.51i·31-s + ⋯
L(s)  = 1  + (−0.215 + 0.976i)3-s + 1.99i·5-s − 0.818i·7-s + (−0.907 − 0.420i)9-s − 0.845·11-s − 1.36·13-s + (−1.94 − 0.429i)15-s − 0.499i·17-s − 0.229i·19-s + (0.799 + 0.176i)21-s + 1.24·23-s − 2.98·25-s + (0.605 − 0.795i)27-s + 1.40i·29-s − 1.16i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.674 + 0.738i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.674 + 0.738i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175600 - 0.398407i\)
\(L(\frac12)\) \(\approx\) \(0.175600 - 0.398407i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.372 - 1.69i)T \)
19 \( 1 + iT \)
good5 \( 1 - 4.46iT - 5T^{2} \)
7 \( 1 + 2.16iT - 7T^{2} \)
11 \( 1 + 2.80T + 11T^{2} \)
13 \( 1 + 4.91T + 13T^{2} \)
17 \( 1 + 2.05iT - 17T^{2} \)
23 \( 1 - 5.97T + 23T^{2} \)
29 \( 1 - 7.58iT - 29T^{2} \)
31 \( 1 + 6.51iT - 31T^{2} \)
37 \( 1 - 0.923T + 37T^{2} \)
41 \( 1 - 4.75iT - 41T^{2} \)
43 \( 1 - 7.87iT - 43T^{2} \)
47 \( 1 + 1.70T + 47T^{2} \)
53 \( 1 + 1.41iT - 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 1.72T + 61T^{2} \)
67 \( 1 + 5.78iT - 67T^{2} \)
71 \( 1 + 4.48T + 71T^{2} \)
73 \( 1 + 3.82T + 73T^{2} \)
79 \( 1 + 1.14iT - 79T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 - 10.6iT - 89T^{2} \)
97 \( 1 + 4.86T + 97T^{2} \)
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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

−10.68233768599876189053868769067, −9.957460508821570485065424365724, −9.377214368893212414121154143401, −7.81217832462030004479123619180, −7.22104268941795558138312640485, −6.45590036477987764449928731408, −5.30424136770606600204094421581, −4.37482100876507845814498055018, −3.14522538904674961333368168278, −2.68549278509381631862740758335, 0.20116798404781424038990131756, 1.60784568415589477662800211412, 2.65001841099679536485256354748, 4.51360466239504760892133558469, 5.33274221764718224643786184440, 5.75479707342293395421266931101, 7.17415392032514450995039356412, 7.998970937454676321040828768993, 8.644619975791268336417299424607, 9.274856801078867490932457514142

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