Properties

Label 2-912-76.71-c1-0-0
Degree $2$
Conductor $912$
Sign $-0.890 + 0.454i$
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  + (−0.173 + 0.984i)3-s + (1.00 + 0.844i)5-s + (−3.15 + 1.82i)7-s + (−0.939 − 0.342i)9-s + (−2.08 − 1.20i)11-s + (−1.01 + 0.179i)13-s + (−1.00 + 0.844i)15-s + (−0.154 + 0.0563i)17-s + (−4.09 − 1.48i)19-s + (−1.24 − 3.42i)21-s + (−1.55 − 1.84i)23-s + (−0.568 − 3.22i)25-s + (0.5 − 0.866i)27-s + (−0.705 + 1.93i)29-s + (0.920 + 1.59i)31-s + ⋯
L(s)  = 1  + (−0.100 + 0.568i)3-s + (0.449 + 0.377i)5-s + (−1.19 + 0.689i)7-s + (−0.313 − 0.114i)9-s + (−0.627 − 0.362i)11-s + (−0.282 + 0.0498i)13-s + (−0.259 + 0.217i)15-s + (−0.0375 + 0.0136i)17-s + (−0.940 − 0.339i)19-s + (−0.272 − 0.748i)21-s + (−0.323 − 0.385i)23-s + (−0.113 − 0.645i)25-s + (0.0962 − 0.166i)27-s + (−0.131 + 0.360i)29-s + (0.165 + 0.286i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $-0.890 + 0.454i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ -0.890 + 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0500661 - 0.208247i\)
\(L(\frac12)\) \(\approx\) \(0.0500661 - 0.208247i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.173 - 0.984i)T \)
19 \( 1 + (4.09 + 1.48i)T \)
good5 \( 1 + (-1.00 - 0.844i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (3.15 - 1.82i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.08 + 1.20i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.01 - 0.179i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (0.154 - 0.0563i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (1.55 + 1.84i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (0.705 - 1.93i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-0.920 - 1.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.17iT - 37T^{2} \)
41 \( 1 + (4.45 + 0.785i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.789 - 0.940i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (0.0928 - 0.254i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (-3.02 - 3.60i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (7.95 - 2.89i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-1.18 + 0.996i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-3.43 - 1.25i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (4.24 + 3.56i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.335 + 1.90i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.63 - 9.27i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (15.2 - 8.81i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (16.9 - 2.98i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (5.59 + 15.3i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.38411971847576878194577925301, −9.863425680858491137122736668244, −9.010963676878085921467595164380, −8.291490215567223684432793432457, −6.94348824272816846589350543492, −6.19390006913328142592208501731, −5.50000638846212340974458427800, −4.34118459017397547018840236928, −3.11960551307043505270160680376, −2.37945419904969704001271041495, 0.094246628205369093347940660990, 1.75590626802291516050102284863, 3.00249671054231936167214564214, 4.17503162007149361947625921034, 5.36577063842812827991892709605, 6.23188755702099962429432109611, 7.01024948060044049840281682343, 7.80436135925124115133140219550, 8.787498198417034155220550906949, 9.819926249091950476811279093056

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