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Properties

Label 2-102-51.38-c4-0-11
Degree $2$
Conductor $102$
Sign $0.354 - 0.934i$
Analytic cond. $10.5437$
Root an. cond. $3.24711$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82·2-s + (4.63 + 7.71i)3-s + 8.00·4-s + (1.85 + 1.85i)5-s + (13.1 + 21.8i)6-s + (25.3 + 25.3i)7-s + 22.6·8-s + (−38.0 + 71.5i)9-s + (5.24 + 5.24i)10-s + (14.3 − 14.3i)11-s + (37.0 + 61.7i)12-s + 33.5·13-s + (71.5 + 71.5i)14-s + (−5.71 + 22.9i)15-s + 64.0·16-s + (1.22 + 288. i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.514 + 0.857i)3-s + 0.500·4-s + (0.0742 + 0.0742i)5-s + (0.364 + 0.606i)6-s + (0.516 + 0.516i)7-s + 0.353·8-s + (−0.469 + 0.882i)9-s + (0.0524 + 0.0524i)10-s + (0.118 − 0.118i)11-s + (0.257 + 0.428i)12-s + 0.198·13-s + (0.365 + 0.365i)14-s + (−0.0254 + 0.101i)15-s + 0.250·16-s + (0.00422 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $0.354 - 0.934i$
Analytic conductor: \(10.5437\)
Root analytic conductor: \(3.24711\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{102} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :2),\ 0.354 - 0.934i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.63135 + 1.81561i\)
\(L(\frac12)\) \(\approx\) \(2.63135 + 1.81561i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82T \)
3 \( 1 + (-4.63 - 7.71i)T \)
17 \( 1 + (-1.22 - 288. i)T \)
good5 \( 1 + (-1.85 - 1.85i)T + 625iT^{2} \)
7 \( 1 + (-25.3 - 25.3i)T + 2.40e3iT^{2} \)
11 \( 1 + (-14.3 + 14.3i)T - 1.46e4iT^{2} \)
13 \( 1 - 33.5T + 2.85e4T^{2} \)
19 \( 1 + 149. iT - 1.30e5T^{2} \)
23 \( 1 + (219. - 219. i)T - 2.79e5iT^{2} \)
29 \( 1 + (-482. - 482. i)T + 7.07e5iT^{2} \)
31 \( 1 + (-360. + 360. i)T - 9.23e5iT^{2} \)
37 \( 1 + (-979. + 979. i)T - 1.87e6iT^{2} \)
41 \( 1 + (-733. + 733. i)T - 2.82e6iT^{2} \)
43 \( 1 + 904. iT - 3.41e6T^{2} \)
47 \( 1 + 2.29e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.52e3T + 7.89e6T^{2} \)
59 \( 1 + 1.14e3T + 1.21e7T^{2} \)
61 \( 1 + (2.72e3 + 2.72e3i)T + 1.38e7iT^{2} \)
67 \( 1 + 423.T + 2.01e7T^{2} \)
71 \( 1 + (2.73e3 + 2.73e3i)T + 2.54e7iT^{2} \)
73 \( 1 + (-6.31e3 + 6.31e3i)T - 2.83e7iT^{2} \)
79 \( 1 + (-2.56e3 - 2.56e3i)T + 3.89e7iT^{2} \)
83 \( 1 + 6.29e3T + 4.74e7T^{2} \)
89 \( 1 - 5.58e3iT - 6.27e7T^{2} \)
97 \( 1 + (-4.09e3 + 4.09e3i)T - 8.85e7iT^{2} \)
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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

−13.56502020697497155895309082102, −12.29178585649657023593745816488, −11.17479101666809987596879931350, −10.25497022437421992313976296294, −8.917447901574132471818829896329, −7.911382472183721449904614058610, −6.14027536958091885083726620643, −4.92149277761033176694459989159, −3.72450741190205292958582278597, −2.23524348353662574730107971033, 1.27910770688298824162811283191, 2.89359120068375579273825633671, 4.47225680595630291266937213663, 6.06806209418210901083718911469, 7.24521645527625344208900292490, 8.175204384681687819901440851550, 9.616075008077729508005333994783, 11.14092476897954210337630167278, 12.05628378075574440253077681701, 13.07453176983598144841069245842

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