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Properties

Label 2-912-12.11-c1-0-3
Degree $2$
Conductor $912$
Sign $-0.301 - 0.953i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.301 - 0.953i$
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: \((2,\ 912,\ (\ :1/2),\ -0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.659238 + 0.900266i\)
\(L(\frac12)\) \(\approx\) \(0.659238 + 0.900266i\)
\(L(\frac{3}{2})\) 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

−10.33612551214596481009799225654, −9.565565274319222573114324810010, −8.567693082650020610722221184320, −7.49843051204279149777518646593, −6.97968681826530678966787423239, −6.34196578713904267954065045541, −5.40697715124304249732169417999, −3.63521496553433776484724331684, −2.73233378013562329344269144004, −2.01862519352660260064211650922, 0.48598480121920988440994642086, 2.10191230251820063086186800799, 4.05795811054696070182531077081, 4.24201599105390170286558232252, 5.20602780735725988239323294850, 6.10700075701818038558540782516, 7.71053535249423060040501712675, 8.213008843647718320320266927748, 9.301775576780206082723429789712, 9.577441514753435822022055471394

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