Properties

Label 2-912-912.341-c1-0-87
Degree $2$
Conductor $912$
Sign $-0.386 + 0.922i$
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  + (0.0274 − 1.41i)2-s + (−0.321 + 1.70i)3-s + (−1.99 − 0.0776i)4-s + (−1.52 + 1.52i)5-s + (2.39 + 0.501i)6-s − 0.618i·7-s + (−0.164 + 2.82i)8-s + (−2.79 − 1.09i)9-s + (2.11 + 2.20i)10-s + (0.333 − 0.333i)11-s + (0.775 − 3.37i)12-s + (−3.67 − 3.67i)13-s + (−0.874 − 0.0169i)14-s + (−2.10 − 3.08i)15-s + (3.98 + 0.310i)16-s + 3.31i·17-s + ⋯
L(s)  = 1  + (0.0194 − 0.999i)2-s + (−0.185 + 0.982i)3-s + (−0.999 − 0.0388i)4-s + (−0.682 + 0.682i)5-s + (0.978 + 0.204i)6-s − 0.233i·7-s + (−0.0582 + 0.998i)8-s + (−0.930 − 0.365i)9-s + (0.669 + 0.695i)10-s + (0.100 − 0.100i)11-s + (0.223 − 0.974i)12-s + (−1.01 − 1.01i)13-s + (−0.233 − 0.00453i)14-s + (−0.543 − 0.797i)15-s + (0.996 + 0.0775i)16-s + 0.802i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.347424 - 0.522464i\)
\(L(\frac12)\) \(\approx\) \(0.347424 - 0.522464i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0274 + 1.41i)T \)
3 \( 1 + (0.321 - 1.70i)T \)
19 \( 1 + (-4.21 + 1.12i)T \)
good5 \( 1 + (1.52 - 1.52i)T - 5iT^{2} \)
7 \( 1 + 0.618iT - 7T^{2} \)
11 \( 1 + (-0.333 + 0.333i)T - 11iT^{2} \)
13 \( 1 + (3.67 + 3.67i)T + 13iT^{2} \)
17 \( 1 - 3.31iT - 17T^{2} \)
23 \( 1 + 2.68T + 23T^{2} \)
29 \( 1 + (-7.04 + 7.04i)T - 29iT^{2} \)
31 \( 1 + 6.61iT - 31T^{2} \)
37 \( 1 + (-2.95 + 2.95i)T - 37iT^{2} \)
41 \( 1 + 3.83iT - 41T^{2} \)
43 \( 1 + (4.64 + 4.64i)T + 43iT^{2} \)
47 \( 1 - 4.73iT - 47T^{2} \)
53 \( 1 + (6.73 + 6.73i)T + 53iT^{2} \)
59 \( 1 + (8.83 + 8.83i)T + 59iT^{2} \)
61 \( 1 + (0.962 - 0.962i)T - 61iT^{2} \)
67 \( 1 + (-0.863 - 0.863i)T + 67iT^{2} \)
71 \( 1 + 2.51iT - 71T^{2} \)
73 \( 1 + 0.167iT - 73T^{2} \)
79 \( 1 - 8.48iT - 79T^{2} \)
83 \( 1 + (2.74 + 2.74i)T + 83iT^{2} \)
89 \( 1 + 3.92iT - 89T^{2} \)
97 \( 1 - 5.42iT - 97T^{2} \)
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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

−10.03254461116338483489090118040, −9.416420691225087959210124670437, −8.228683645473783335214280817827, −7.61713908994920960595315993673, −6.09682480749327213550398981494, −5.13197977834109965205050691742, −4.18668835818731977989693081170, −3.42560083559447879384523644549, −2.55582704205119516455915442985, −0.34788553868949039926649016256, 1.21071743729630697930672814666, 3.01717633355282361171816559410, 4.58598242332734195957342300752, 5.08408304590077217251624758960, 6.25850057819676499172693160002, 7.06550676683198304603201802466, 7.64723525552053088034459997615, 8.501156925194978516734080943368, 9.128879203801848036485868451202, 10.10290146933226831790931027890

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