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Properties

Label 2-912-76.67-c1-0-11
Degree $2$
Conductor $912$
Sign $0.945 - 0.325i$
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  + (−0.766 + 0.642i)3-s + (1.72 + 0.628i)5-s + (3.53 + 2.03i)7-s + (0.173 − 0.984i)9-s + (4.09 − 2.36i)11-s + (2.35 − 2.80i)13-s + (−1.72 + 0.628i)15-s + (−1.27 − 7.22i)17-s + (−3.86 + 2.02i)19-s + (−4.01 + 0.707i)21-s + (1.09 + 3.00i)23-s + (−1.24 − 1.04i)25-s + (0.500 + 0.866i)27-s + (2.75 + 0.485i)29-s + (−3.49 + 6.04i)31-s + ⋯
L(s)  = 1  + (−0.442 + 0.371i)3-s + (0.772 + 0.281i)5-s + (1.33 + 0.770i)7-s + (0.0578 − 0.328i)9-s + (1.23 − 0.713i)11-s + (0.651 − 0.776i)13-s + (−0.445 + 0.162i)15-s + (−0.308 − 1.75i)17-s + (−0.885 + 0.464i)19-s + (−0.876 + 0.154i)21-s + (0.228 + 0.626i)23-s + (−0.248 − 0.208i)25-s + (0.0962 + 0.166i)27-s + (0.511 + 0.0901i)29-s + (−0.627 + 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.945 - 0.325i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.945 - 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91171 + 0.319792i\)
\(L(\frac12)\) \(\approx\) \(1.91171 + 0.319792i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.766 - 0.642i)T \)
19 \( 1 + (3.86 - 2.02i)T \)
good5 \( 1 + (-1.72 - 0.628i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-3.53 - 2.03i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.09 + 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.35 + 2.80i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.27 + 7.22i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-1.09 - 3.00i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-2.75 - 0.485i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (3.49 - 6.04i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.49iT - 37T^{2} \)
41 \( 1 + (-3.61 - 4.30i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (-2.83 + 7.79i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-2.50 - 0.441i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-1.47 - 4.05i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.29 + 7.32i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (1.56 - 0.569i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.27 - 7.23i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (10.8 + 3.94i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (8.89 - 7.46i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (4.80 - 4.02i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-9.06 - 5.23i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.28 + 3.90i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (18.6 - 3.28i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.27232723884750637000232573548, −9.075765057336865047922781725184, −8.814837105761202148251615809078, −7.61993807833498634766277515938, −6.46881961926909818009008544338, −5.71033536914710934617688224318, −5.06201079933715407532279457228, −3.89185231934088632171338332357, −2.56207369024326169390620233179, −1.26912819691977233399414555272, 1.41022104325625144033477023047, 1.90449024600470849824631944562, 4.19763857297997372486934228861, 4.47261779479777634126805434965, 5.91494510930908214189849559657, 6.49418141441809084307514525158, 7.43261137626948969047575583744, 8.443310236552990831001147557333, 9.099693513839584683492282600524, 10.20177676610590370305014515073

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