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Properties

Label 2-912-304.85-c1-0-79
Degree $2$
Conductor $912$
Sign $-0.829 - 0.558i$
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  + (0.980 − 1.01i)2-s + (0.0871 − 0.996i)3-s + (−0.0782 − 1.99i)4-s + (−0.978 − 2.09i)5-s + (−0.930 − 1.06i)6-s + (−2.70 − 1.56i)7-s + (−2.11 − 1.87i)8-s + (−0.984 − 0.173i)9-s + (−3.09 − 1.05i)10-s + (−0.184 + 0.687i)11-s + (−1.99 − 0.0962i)12-s + (5.61 − 0.490i)13-s + (−4.24 + 1.22i)14-s + (−2.17 + 0.792i)15-s + (−3.98 + 0.312i)16-s + (−0.125 − 0.714i)17-s + ⋯
L(s)  = 1  + (0.693 − 0.720i)2-s + (0.0503 − 0.575i)3-s + (−0.0391 − 0.999i)4-s + (−0.437 − 0.938i)5-s + (−0.379 − 0.434i)6-s + (−1.02 − 0.590i)7-s + (−0.747 − 0.664i)8-s + (−0.328 − 0.0578i)9-s + (−0.979 − 0.335i)10-s + (−0.0555 + 0.207i)11-s + (−0.576 − 0.0277i)12-s + (1.55 − 0.136i)13-s + (−1.13 + 0.327i)14-s + (−0.561 + 0.204i)15-s + (−0.996 + 0.0781i)16-s + (−0.0305 − 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.829 - 0.558i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.829 - 0.558i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433880 + 1.42031i\)
\(L(\frac12)\) \(\approx\) \(0.433880 + 1.42031i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.980 + 1.01i)T \)
3 \( 1 + (-0.0871 + 0.996i)T \)
19 \( 1 + (0.896 - 4.26i)T \)
good5 \( 1 + (0.978 + 2.09i)T + (-3.21 + 3.83i)T^{2} \)
7 \( 1 + (2.70 + 1.56i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.184 - 0.687i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-5.61 + 0.490i)T + (12.8 - 2.25i)T^{2} \)
17 \( 1 + (0.125 + 0.714i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-0.559 - 1.53i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.669 - 0.956i)T + (-9.91 + 27.2i)T^{2} \)
31 \( 1 + (-0.750 + 1.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.25 + 1.25i)T + 37iT^{2} \)
41 \( 1 + (3.03 + 3.61i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (2.24 + 4.81i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (0.329 - 1.86i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (9.02 + 4.20i)T + (34.0 + 40.6i)T^{2} \)
59 \( 1 + (1.59 - 2.27i)T + (-20.1 - 55.4i)T^{2} \)
61 \( 1 + (-4.64 + 9.95i)T + (-39.2 - 46.7i)T^{2} \)
67 \( 1 + (0.0812 + 0.116i)T + (-22.9 + 62.9i)T^{2} \)
71 \( 1 + (-4.72 + 12.9i)T + (-54.3 - 45.6i)T^{2} \)
73 \( 1 + (-3.96 - 4.72i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-8.61 + 7.22i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.126 - 0.471i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-1.16 + 1.38i)T + (-15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.0799 - 0.453i)T + (-91.1 + 33.1i)T^{2} \)
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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

−9.704146791812054762408475348203, −8.838767533365301364171157746438, −8.003013371648692761590617506035, −6.75572275060845902239658775542, −6.09598655131403828284970434413, −5.06992442880146109915508593150, −3.90393191879044099485685501499, −3.33927793871227108666621904140, −1.70503015938222361333711951840, −0.55986308475453680155010764929, 2.81313543561116603232792293005, 3.35894179316740068179986861403, 4.28141439577954869745922266734, 5.48994919843575846977284616768, 6.43761304078421309943032262310, 6.76819807577248956769454870463, 8.087907903394908127550011900660, 8.773923903616657317195329665374, 9.601642482653993786767205191590, 10.83978966291028256489946373210

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