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Properties

Label 2-912-57.14-c1-0-36
Degree $2$
Conductor $912$
Sign $-0.951 - 0.308i$
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  + (−0.779 − 1.54i)3-s + (1.86 − 2.21i)5-s + (−0.562 − 0.973i)7-s + (−1.78 + 2.41i)9-s + (−2.70 − 1.56i)11-s + (−5.18 + 0.914i)13-s + (−4.88 − 1.14i)15-s + (−0.880 − 2.41i)17-s + (−4.13 + 1.37i)19-s + (−1.06 + 1.62i)21-s + (4.31 + 5.14i)23-s + (−0.589 − 3.34i)25-s + (5.12 + 0.876i)27-s + (−1.09 − 0.399i)29-s + (3.90 − 2.25i)31-s + ⋯
L(s)  = 1  + (−0.450 − 0.892i)3-s + (0.832 − 0.992i)5-s + (−0.212 − 0.367i)7-s + (−0.594 + 0.804i)9-s + (−0.816 − 0.471i)11-s + (−1.43 + 0.253i)13-s + (−1.26 − 0.296i)15-s + (−0.213 − 0.586i)17-s + (−0.948 + 0.315i)19-s + (−0.232 + 0.355i)21-s + (0.900 + 1.07i)23-s + (−0.117 − 0.668i)25-s + (0.985 + 0.168i)27-s + (−0.203 − 0.0740i)29-s + (0.701 − 0.405i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 - 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.951 - 0.308i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.951 - 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101573 + 0.643373i\)
\(L(\frac12)\) \(\approx\) \(0.101573 + 0.643373i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.779 + 1.54i)T \)
19 \( 1 + (4.13 - 1.37i)T \)
good5 \( 1 + (-1.86 + 2.21i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (0.562 + 0.973i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.70 + 1.56i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.18 - 0.914i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.880 + 2.41i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-4.31 - 5.14i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.09 + 0.399i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.90 + 2.25i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 12.0iT - 37T^{2} \)
41 \( 1 + (1.06 - 6.02i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (2.21 + 1.85i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (0.377 - 1.03i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (5.66 - 4.75i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (6.41 - 2.33i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (5.58 - 4.68i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-2.42 + 6.64i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (3.31 + 2.77i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (1.30 - 7.41i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (4.30 + 0.759i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-12.5 + 7.22i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.38 + 13.5i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (1.47 + 4.04i)T + (-74.3 + 62.3i)T^{2} \)
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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

−9.569182349969049018754016410251, −8.841978227798654398625404327947, −7.77561041477864281296678885056, −7.15525390294119810543223039539, −6.05588394633496091344025930522, −5.32909302289015097586514432997, −4.60334511593513180486613693572, −2.76536025959659740722190491305, −1.72972838733952032936369964582, −0.30054079552067851553016602917, 2.38262052280476092230427206424, 3.02159682022448467696561083938, 4.58860293266890445335350184229, 5.20068343884718430989818319012, 6.33808850632270034964302259245, 6.79673542738354232363669467557, 8.124102915019877372707149376527, 9.157538679459480112189630092457, 9.987396567399655729448843117599, 10.41587302059381273599176393889

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