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Properties

Label 2-798-399.179-c1-0-36
Degree $2$
Conductor $798$
Sign $0.991 - 0.127i$
Analytic cond. $6.37206$
Root an. cond. $2.52429$
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  + 2-s + (1.69 − 0.373i)3-s + 4-s + 3.13i·5-s + (1.69 − 0.373i)6-s + (−0.151 − 2.64i)7-s + 8-s + (2.72 − 1.26i)9-s + 3.13i·10-s + (−1.01 − 0.584i)11-s + (1.69 − 0.373i)12-s + (3.50 − 2.02i)13-s + (−0.151 − 2.64i)14-s + (1.17 + 5.30i)15-s + 16-s + (−0.268 + 0.154i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.976 − 0.215i)3-s + 0.5·4-s + 1.40i·5-s + (0.690 − 0.152i)6-s + (−0.0573 − 0.998i)7-s + 0.353·8-s + (0.907 − 0.421i)9-s + 0.991i·10-s + (−0.305 − 0.176i)11-s + (0.488 − 0.107i)12-s + (0.972 − 0.561i)13-s + (−0.0405 − 0.705i)14-s + (0.302 + 1.36i)15-s + 0.250·16-s + (−0.0650 + 0.0375i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(798\)    =    \(2 \cdot 3 \cdot 7 \cdot 19\)
Sign: $0.991 - 0.127i$
Analytic conductor: \(6.37206\)
Root analytic conductor: \(2.52429\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{798} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 798,\ (\ :1/2),\ 0.991 - 0.127i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.32608 + 0.213113i\)
\(L(\frac12)\) \(\approx\) \(3.32608 + 0.213113i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.69 + 0.373i)T \)
7 \( 1 + (0.151 + 2.64i)T \)
19 \( 1 + (0.785 - 4.28i)T \)
good5 \( 1 - 3.13iT - 5T^{2} \)
11 \( 1 + (1.01 + 0.584i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.50 + 2.02i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.268 - 0.154i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.63 + 0.945i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.886 - 1.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.83 - 2.21i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.66 - 2.69i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.874 - 1.51i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.39 - 7.61i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.560 + 0.323i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 + (7.31 + 12.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.82 - 3.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + 14.7iT - 67T^{2} \)
71 \( 1 + (-2.03 + 3.51i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.96 + 3.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 8.95iT - 79T^{2} \)
83 \( 1 - 4.89iT - 83T^{2} \)
89 \( 1 + (-4.92 + 8.53i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.764 - 0.441i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.55808632440230259703271694146, −9.663138752626346121633297227710, −8.167590096305204981664011520159, −7.75928321079235667019042708677, −6.65641118418172265235022136717, −6.27738346551026101099217277894, −4.61904117812856183305413041166, −3.37608902551157965405274232565, −3.21519202223927507427175083463, −1.67903673533545776466640160933, 1.61430078076703167930279721855, 2.69143149599617096778465931569, 3.95506200474586153908502644935, 4.75141472976300564178983087182, 5.55677404742659944727858602560, 6.69225193249800760851625491795, 7.918008859837002783023480124375, 8.727727886321659662189536279731, 9.081470909915817619499430731749, 10.09158782226819628550925225517

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