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Properties

Label 2-312-104.101-c1-0-14
Degree $2$
Conductor $312$
Sign $0.987 - 0.157i$
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  + (−1.17 + 0.783i)2-s + (0.866 − 0.5i)3-s + (0.771 − 1.84i)4-s + 3.08·5-s + (−0.627 + 1.26i)6-s + (0.257 + 0.148i)7-s + (0.538 + 2.77i)8-s + (0.499 − 0.866i)9-s + (−3.63 + 2.41i)10-s + (0.903 + 1.56i)11-s + (−0.254 − 1.98i)12-s + (−3.08 − 1.86i)13-s + (−0.419 + 0.0268i)14-s + (2.67 − 1.54i)15-s + (−2.81 − 2.84i)16-s + (3.83 − 6.64i)17-s + ⋯
L(s)  = 1  + (−0.832 + 0.554i)2-s + (0.499 − 0.288i)3-s + (0.385 − 0.922i)4-s + 1.37·5-s + (−0.256 + 0.517i)6-s + (0.0973 + 0.0562i)7-s + (0.190 + 0.981i)8-s + (0.166 − 0.288i)9-s + (−1.14 + 0.764i)10-s + (0.272 + 0.471i)11-s + (−0.0735 − 0.572i)12-s + (−0.855 − 0.518i)13-s + (−0.112 + 0.00717i)14-s + (0.689 − 0.398i)15-s + (−0.702 − 0.711i)16-s + (0.930 − 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.987 - 0.157i$
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: \((2,\ 312,\ (\ :1/2),\ 0.987 - 0.157i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29875 + 0.102646i\)
\(L(\frac12)\) \(\approx\) \(1.29875 + 0.102646i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 - 0.783i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (3.08 + 1.86i)T \)
good5 \( 1 - 3.08T + 5T^{2} \)
7 \( 1 + (-0.257 - 0.148i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.903 - 1.56i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.83 + 6.64i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.64 - 2.84i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.85 - 4.95i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.98 - 2.87i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.26iT - 31T^{2} \)
37 \( 1 + (-1.78 - 3.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.43 + 3.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.92 + 1.68i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.73iT - 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 + (-3.79 + 6.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.60 + 3.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.96 - 6.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.3 + 5.95i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.76iT - 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 + 9.73T + 83T^{2} \)
89 \( 1 + (5.44 - 3.14i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.86 - 3.96i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.58255152494417537921191075636, −10.28142148734099467359273420669, −9.593915887962465373984046286877, −9.141994477749573196180315198498, −7.73933922897501885000657733473, −7.12219775080610509561718667471, −5.88615970243333775061173636253, −5.09226554435671340275346641708, −2.73685573255764968119651179368, −1.51895205906674376382321273233, 1.69059720109703493450371096159, 2.77893508667426046499501437265, 4.23770400610707733054353712344, 5.84617344851210402553708982412, 6.95300973041209368446903230729, 8.210006798006739210465192112855, 9.031445483899546342890807536443, 9.799894558732093428795281445741, 10.42506027925689726794549054662, 11.37002550985245806229256693719

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