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Properties

Label 2-912-76.51-c1-0-19
Degree $2$
Conductor $912$
Sign $-0.999 - 0.0209i$
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  + (0.939 + 0.342i)3-s + (−0.749 − 4.25i)5-s + (−3.63 − 2.09i)7-s + (0.766 + 0.642i)9-s + (−3.21 + 1.85i)11-s + (1.64 + 4.52i)13-s + (0.749 − 4.25i)15-s + (0.869 − 0.729i)17-s + (−4.06 − 1.57i)19-s + (−2.69 − 3.21i)21-s + (0.795 + 0.140i)23-s + (−12.8 + 4.66i)25-s + (0.500 + 0.866i)27-s + (2.83 − 3.37i)29-s + (−2.66 + 4.61i)31-s + ⋯
L(s)  = 1  + (0.542 + 0.197i)3-s + (−0.335 − 1.90i)5-s + (−1.37 − 0.792i)7-s + (0.255 + 0.214i)9-s + (−0.968 + 0.559i)11-s + (0.457 + 1.25i)13-s + (0.193 − 1.09i)15-s + (0.210 − 0.176i)17-s + (−0.932 − 0.361i)19-s + (−0.588 − 0.700i)21-s + (0.165 + 0.0292i)23-s + (−2.56 + 0.933i)25-s + (0.0962 + 0.166i)27-s + (0.526 − 0.627i)29-s + (−0.478 + 0.829i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0209i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.999 - 0.0209i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.999 - 0.0209i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00557055 + 0.530504i\)
\(L(\frac12)\) \(\approx\) \(0.00557055 + 0.530504i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.939 - 0.342i)T \)
19 \( 1 + (4.06 + 1.57i)T \)
good5 \( 1 + (0.749 + 4.25i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (3.63 + 2.09i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.21 - 1.85i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.64 - 4.52i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.869 + 0.729i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-0.795 - 0.140i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-2.83 + 3.37i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (2.66 - 4.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.90iT - 37T^{2} \)
41 \( 1 + (1.34 - 3.69i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (4.88 - 0.860i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-0.921 + 1.09i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (9.28 + 1.63i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-1.86 + 1.56i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.46 + 13.9i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.583 + 0.489i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (2.17 + 12.3i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (7.65 + 2.78i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (5.89 + 2.14i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-5.90 - 3.40i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.639 + 1.75i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (10.5 + 12.5i)T + (-16.8 + 95.5i)T^{2} \)
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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

−9.524634168200604045989240779550, −8.927599909911147060649705736500, −8.160707899507185540082712939678, −7.24737905810465240964445242738, −6.25968645615168708586968835740, −4.91224534543735000608702298516, −4.34478522568180950149467353118, −3.40406604579429956863598647546, −1.80340510327240565857055078674, −0.21930207734838549589881648597, 2.54370175392310692467677788682, 3.02387903717406894075678338598, 3.70403043990400364263286035753, 5.69985071962474105515471133662, 6.26452043125430193386051965295, 7.09511963709262738387040071516, 7.968830271468409222324221419435, 8.693519967010836660717144417387, 9.992789729006685124654951243058, 10.35051999200299620839418885848

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