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Properties

Label 2-912-76.31-c1-0-0
Degree $2$
Conductor $912$
Sign $-0.907 - 0.420i$
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  + (0.5 + 0.866i)3-s + (−1.73 − 3.01i)5-s + 3.36i·7-s + (−0.499 + 0.866i)9-s + 2.43i·11-s + (−5.02 − 2.89i)13-s + (1.73 − 3.01i)15-s + (2.67 + 4.63i)17-s + (−2.84 − 3.30i)19-s + (−2.91 + 1.68i)21-s + (1.30 + 0.753i)23-s + (−3.54 + 6.13i)25-s − 0.999·27-s + (−4.21 − 2.43i)29-s − 10.5·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.777 − 1.34i)5-s + 1.27i·7-s + (−0.166 + 0.288i)9-s + 0.733i·11-s + (−1.39 − 0.804i)13-s + (0.448 − 0.777i)15-s + (0.649 + 1.12i)17-s + (−0.652 − 0.757i)19-s + (−0.636 + 0.367i)21-s + (0.272 + 0.157i)23-s + (−0.708 + 1.22i)25-s − 0.192·27-s + (−0.782 − 0.451i)29-s − 1.89·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.907 - 0.420i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.907 - 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100750 + 0.456698i\)
\(L(\frac12)\) \(\approx\) \(0.100750 + 0.456698i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (2.84 + 3.30i)T \)
good5 \( 1 + (1.73 + 3.01i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.36iT - 7T^{2} \)
11 \( 1 - 2.43iT - 11T^{2} \)
13 \( 1 + (5.02 + 2.89i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.67 - 4.63i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-1.30 - 0.753i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.21 + 2.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 - 8.46iT - 37T^{2} \)
41 \( 1 + (3.41 - 1.96i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.02 - 5.21i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.63 - 5.56i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.41 + 2.54i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.58 - 9.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.80 - 4.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.03 + 5.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.15 + 10.6i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.97 + 6.88i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.19 - 2.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 5.59iT - 83T^{2} \)
89 \( 1 + (5.04 + 2.91i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.47 + 1.43i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.25236494957185361457350478350, −9.438502737278105516540209362387, −8.817841063364063689684262749067, −8.111091236320442301289928767242, −7.37475905844013715947690247638, −5.80216377352910299092601286055, −5.04515697246904866459802611412, −4.41752754411334773883725591576, −3.15891123996725230497675386917, −1.88540509896002475724615135974, 0.20061494322440723420235430499, 2.14690688734284147139537249660, 3.38502962399243807081535213154, 3.94910938185792597093593817693, 5.38797843553571735594192914984, 6.76935861330309067496857511571, 7.28457320603813802686603336454, 7.57586249719345265481480955992, 8.832292030391923744904759565022, 9.885256274349431195943781226558

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