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Properties

Label 2-912-76.51-c1-0-10
Degree $2$
Conductor $912$
Sign $0.433 + 0.901i$
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 + (0.525 + 2.98i)5-s + (−3.04 − 1.75i)7-s + (0.766 + 0.642i)9-s + (−3.50 + 2.02i)11-s + (−1.45 − 3.98i)13-s + (0.525 − 2.98i)15-s + (5.10 − 4.28i)17-s + (−0.436 − 4.33i)19-s + (2.26 + 2.69i)21-s + (7.06 + 1.24i)23-s + (−3.91 + 1.42i)25-s + (−0.500 − 0.866i)27-s + (1.31 − 1.56i)29-s + (2.83 − 4.90i)31-s + ⋯
L(s)  = 1  + (−0.542 − 0.197i)3-s + (0.235 + 1.33i)5-s + (−1.15 − 0.664i)7-s + (0.255 + 0.214i)9-s + (−1.05 + 0.609i)11-s + (−0.402 − 1.10i)13-s + (0.135 − 0.769i)15-s + (1.23 − 1.03i)17-s + (−0.100 − 0.994i)19-s + (0.493 + 0.587i)21-s + (1.47 + 0.259i)23-s + (−0.783 + 0.285i)25-s + (−0.0962 − 0.166i)27-s + (0.243 − 0.290i)29-s + (0.508 − 0.881i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.433 + 0.901i$
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: \((2,\ 912,\ (\ :1/2),\ 0.433 + 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.739149 - 0.464503i\)
\(L(\frac12)\) \(\approx\) \(0.739149 - 0.464503i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.436 + 4.33i)T \)
good5 \( 1 + (-0.525 - 2.98i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (3.04 + 1.75i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.50 - 2.02i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.45 + 3.98i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-5.10 + 4.28i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-7.06 - 1.24i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.31 + 1.56i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.83 + 4.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 9.28iT - 37T^{2} \)
41 \( 1 + (-1.62 + 4.46i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-8.66 + 1.52i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.48 - 1.76i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (10.0 + 1.77i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-1.21 + 1.01i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-2.02 + 11.4i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.41 - 1.18i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.499 + 2.83i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-5.05 - 1.83i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (9.34 + 3.40i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (2.58 + 1.49i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.71 + 15.7i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (7.29 + 8.69i)T + (-16.8 + 95.5i)T^{2} \)
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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

−9.976593195954162210986216975652, −9.623057594318448357826162937631, −7.85817918699492520568945657092, −7.21278755226932930277232927288, −6.70880122275153908528556088185, −5.67557181162155913621674964181, −4.77970604727558128475709743424, −3.09576194998462793314886378970, −2.78354238788956419977451254390, −0.49768785827061488946002068390, 1.20661513246256765988554722058, 2.83557881873791765266394954435, 4.06974370916052969782321258290, 5.21505030916847579456654215015, 5.70575387624159555939600919499, 6.58336288596689880075442874007, 7.85060752211582552444178249923, 8.778178252283482828963353275887, 9.360125899104460893196738711491, 10.14604837567908936455008798106

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