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Properties

Label 2-312-104.101-c1-0-16
Degree $2$
Conductor $312$
Sign $-0.891 + 0.452i$
Analytic cond. $2.49133$
Root an. cond. $1.57839$
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  + (−0.526 − 1.31i)2-s + (−0.866 + 0.5i)3-s + (−1.44 + 1.38i)4-s + 0.112·5-s + (1.11 + 0.873i)6-s + (0.0378 + 0.0218i)7-s + (2.57 + 1.16i)8-s + (0.499 − 0.866i)9-s + (−0.0592 − 0.147i)10-s + (−3.10 − 5.37i)11-s + (0.560 − 1.91i)12-s + (−1.35 − 3.34i)13-s + (0.00874 − 0.0612i)14-s + (−0.0973 + 0.0562i)15-s + (0.176 − 3.99i)16-s + (1.70 − 2.95i)17-s + ⋯
L(s)  = 1  + (−0.372 − 0.928i)2-s + (−0.499 + 0.288i)3-s + (−0.722 + 0.691i)4-s + 0.0502·5-s + (0.454 + 0.356i)6-s + (0.0143 + 0.00826i)7-s + (0.910 + 0.412i)8-s + (0.166 − 0.288i)9-s + (−0.0187 − 0.0466i)10-s + (−0.935 − 1.62i)11-s + (0.161 − 0.554i)12-s + (−0.375 − 0.926i)13-s + (0.00233 − 0.0163i)14-s + (−0.0251 + 0.0145i)15-s + (0.0440 − 0.999i)16-s + (0.413 − 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.891 + 0.452i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ -0.891 + 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.125371 - 0.524168i\)
\(L(\frac12)\) \(\approx\) \(0.125371 - 0.524168i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.526 + 1.31i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (1.35 + 3.34i)T \)
good5 \( 1 - 0.112T + 5T^{2} \)
7 \( 1 + (-0.0378 - 0.0218i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.10 + 5.37i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.70 + 2.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.27 - 5.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.27 + 3.93i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.22 + 3.01i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.66iT - 31T^{2} \)
37 \( 1 + (-5.07 - 8.78i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.01 + 1.16i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.02 + 4.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.65iT - 47T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 + (1.11 - 1.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.56 - 4.94i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.26 - 3.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.64 - 1.52i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.71iT - 73T^{2} \)
79 \( 1 - 8.01T + 79T^{2} \)
83 \( 1 + 2.86T + 83T^{2} \)
89 \( 1 + (-3.35 + 1.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.971 + 0.560i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.28994949336477538340309800578, −10.21125474639334071427124768809, −9.944837012642344245183030438326, −8.320936324397969610261022586447, −7.979821637674119541355896036132, −6.10979353208795947038094855920, −5.14795271606072417343205454042, −3.79067698837524359695216255782, −2.61365918781970216787180692625, −0.46218826664749537718749047523, 1.91819902356141475979225941643, 4.40526283734641056082703384922, 5.18121108708642070087461024069, 6.43316872271069209682406107237, 7.20753382204541957182931159018, 8.011975279657189718443616076325, 9.271440898648781034790177108355, 10.05751859295382170122314247252, 10.93047420171707220429753632061, 12.24601334259221896514981812829

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