Properties

Label 2-912-57.50-c1-0-1
Degree $2$
Conductor $912$
Sign $-0.217 - 0.976i$
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  + (0.158 − 1.72i)3-s + (−1.22 + 0.707i)5-s + 0.267·7-s + (−2.94 − 0.548i)9-s + 5.27i·11-s + (−0.232 − 0.133i)13-s + (1.02 + 2.22i)15-s + (−4.24 + 2.44i)17-s + (−1.73 − 4i)19-s + (0.0425 − 0.462i)21-s + (−4.57 − 2.63i)23-s + (−1.50 + 2.59i)25-s + (−1.41 + 4.99i)27-s + (−1.03 + 1.79i)29-s + 2.46i·31-s + ⋯
L(s)  = 1  + (0.0917 − 0.995i)3-s + (−0.547 + 0.316i)5-s + 0.101·7-s + (−0.983 − 0.182i)9-s + 1.59i·11-s + (−0.0643 − 0.0371i)13-s + (0.264 + 0.574i)15-s + (−1.02 + 0.594i)17-s + (−0.397 − 0.917i)19-s + (0.00929 − 0.100i)21-s + (−0.953 − 0.550i)23-s + (−0.300 + 0.519i)25-s + (−0.272 + 0.962i)27-s + (−0.192 + 0.332i)29-s + 0.442i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.217 - 0.976i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.217 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.343468 + 0.428259i\)
\(L(\frac12)\) \(\approx\) \(0.343468 + 0.428259i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.158 + 1.72i)T \)
19 \( 1 + (1.73 + 4i)T \)
good5 \( 1 + (1.22 - 0.707i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.267T + 7T^{2} \)
11 \( 1 - 5.27iT - 11T^{2} \)
13 \( 1 + (0.232 + 0.133i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.24 - 2.44i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.57 + 2.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.03 - 1.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.46iT - 31T^{2} \)
37 \( 1 - 7.73iT - 37T^{2} \)
41 \( 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.86 - 4.96i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.656 + 0.378i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.46 + 9.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.60 - 9.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.23 + 9.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.69 + 11.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.06 - 5.23i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.07iT - 83T^{2} \)
89 \( 1 + (3.67 - 6.36i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.464 - 0.267i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.39752056417284564200039685411, −9.413153470421490451716819562831, −8.446711060340473634434965915963, −7.74168347229201480685589194695, −6.89889128135846713628218821821, −6.41912833207372144341487067994, −5.01056250342754943947983008074, −4.06418091952156900545821215912, −2.67288254076284268713814859926, −1.72128082924341836682443571602, 0.24399902680607335928413778972, 2.44057478019176540824317934035, 3.75194825977251030991673795239, 4.22105392031515257806602072715, 5.50644505198221759450641373105, 6.08972356542199230241496514557, 7.55101064028644912551552740674, 8.426418549138449507070944013397, 8.893632250467615455628010731437, 9.865823949964492969621253480271

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