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Properties

Label 2-912-76.71-c1-0-4
Degree $2$
Conductor $912$
Sign $-0.326 - 0.945i$
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  + (−0.173 + 0.984i)3-s + (2.83 + 2.37i)5-s + (−2.26 + 1.30i)7-s + (−0.939 − 0.342i)9-s + (0.444 + 0.256i)11-s + (5.82 − 1.02i)13-s + (−2.83 + 2.37i)15-s + (−3.47 + 1.26i)17-s + (2.31 + 3.69i)19-s + (−0.893 − 2.45i)21-s + (1.12 + 1.34i)23-s + (1.50 + 8.53i)25-s + (0.5 − 0.866i)27-s + (−1.62 + 4.45i)29-s + (−4.54 − 7.87i)31-s + ⋯
L(s)  = 1  + (−0.100 + 0.568i)3-s + (1.26 + 1.06i)5-s + (−0.855 + 0.493i)7-s + (−0.313 − 0.114i)9-s + (0.134 + 0.0774i)11-s + (1.61 − 0.284i)13-s + (−0.731 + 0.613i)15-s + (−0.843 + 0.307i)17-s + (0.532 + 0.846i)19-s + (−0.195 − 0.535i)21-s + (0.234 + 0.279i)23-s + (0.301 + 1.70i)25-s + (0.0962 − 0.166i)27-s + (−0.301 + 0.827i)29-s + (−0.816 − 1.41i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.326 - 0.945i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.984873 + 1.38197i\)
\(L(\frac12)\) \(\approx\) \(0.984873 + 1.38197i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (-2.31 - 3.69i)T \)
good5 \( 1 + (-2.83 - 2.37i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (2.26 - 1.30i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.444 - 0.256i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-5.82 + 1.02i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (3.47 - 1.26i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-1.12 - 1.34i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.62 - 4.45i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.54 + 7.87i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.53iT - 37T^{2} \)
41 \( 1 + (-1.13 - 0.200i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (2.50 - 2.97i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (1.56 - 4.29i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (4.15 + 4.95i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-3.14 + 1.14i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (4.68 - 3.92i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (3.30 + 1.20i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-1.45 - 1.22i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-2.32 + 13.2i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.21 - 6.91i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-12.6 + 7.30i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-15.6 + 2.75i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (0.687 + 1.88i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.47262357359970577479335652119, −9.321933110734528291367262031438, −9.226134884289147111494711669141, −7.82170331677538429153638204239, −6.48370658926425076192686934816, −6.15230417104250130597949123560, −5.39709606190107386777572719946, −3.78041590519314569384929509160, −3.06555608110830367714452421109, −1.84215918807774164847718353128, 0.839332840279782550439219896923, 1.91375420997395028909550144153, 3.32810711894204227721772549150, 4.63229508435859381593502764964, 5.62433084738983355426852708321, 6.40749022136983026878059328388, 7.00284346950858333512690578871, 8.440951509277109706415780386895, 9.002255905135846884559533064283, 9.652182120566799204248305371690

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