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Properties

Label 2-912-76.67-c1-0-3
Degree $2$
Conductor $912$
Sign $0.105 - 0.994i$
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 + (−0.800 − 0.291i)5-s + (−1.70 − 0.983i)7-s + (0.173 − 0.984i)9-s + (−0.742 + 0.428i)11-s + (2.79 − 3.33i)13-s + (0.800 − 0.291i)15-s + (1.26 + 7.14i)17-s + (0.199 + 4.35i)19-s + (1.93 − 0.341i)21-s + (0.620 + 1.70i)23-s + (−3.27 − 2.74i)25-s + (0.500 + 0.866i)27-s + (9.99 + 1.76i)29-s + (−1.88 + 3.25i)31-s + ⋯
L(s)  = 1  + (−0.442 + 0.371i)3-s + (−0.357 − 0.130i)5-s + (−0.644 − 0.371i)7-s + (0.0578 − 0.328i)9-s + (−0.223 + 0.129i)11-s + (0.776 − 0.925i)13-s + (0.206 − 0.0752i)15-s + (0.305 + 1.73i)17-s + (0.0456 + 0.998i)19-s + (0.422 − 0.0745i)21-s + (0.129 + 0.355i)23-s + (−0.654 − 0.549i)25-s + (0.0962 + 0.166i)27-s + (1.85 + 0.327i)29-s + (−0.337 + 0.584i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.105 - 0.994i$
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: \((2,\ 912,\ (\ :1/2),\ 0.105 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.687113 + 0.618100i\)
\(L(\frac12)\) \(\approx\) \(0.687113 + 0.618100i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.199 - 4.35i)T \)
good5 \( 1 + (0.800 + 0.291i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (1.70 + 0.983i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.742 - 0.428i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.79 + 3.33i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-1.26 - 7.14i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.620 - 1.70i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-9.99 - 1.76i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.88 - 3.25i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.42iT - 37T^{2} \)
41 \( 1 + (0.985 + 1.17i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.98 - 10.9i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (7.46 + 1.31i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (-1.86 - 5.11i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (-1.02 - 5.83i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-2.62 + 0.955i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-1.41 + 8.04i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-11.8 - 4.30i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-4.45 + 3.73i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (3.58 - 3.01i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-11.1 - 6.44i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.05 - 7.22i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (7.66 - 1.35i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.26939123583482334380168214443, −9.788892239955219806904744707343, −8.286853189638245719335579386061, −8.115182373053377868337410524369, −6.61282088469448289110422593524, −6.06888110507270350865572098237, −5.01490982984425796481948406540, −3.88825728763229181934696019243, −3.22219679348613531033916775148, −1.26234921495145204647325244141, 0.52065191789826119525756846270, 2.31154488881688255084508333333, 3.41158378865311999503254874715, 4.65393914255315885910810388603, 5.58274697442588578206438783566, 6.63446327142902071703957708989, 7.10204754100457962700738643254, 8.205218519147352961661605592190, 9.131142252492374427014393107197, 9.799206153958535159247097626973

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