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Properties

Label 2-104-104.83-c1-0-8
Degree $2$
Conductor $104$
Sign $0.829 - 0.558i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
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  + (1.15 + 0.813i)2-s + 1.38·3-s + (0.674 + 1.88i)4-s + (−2.40 − 2.40i)5-s + (1.60 + 1.12i)6-s + (−0.127 + 0.127i)7-s + (−0.751 + 2.72i)8-s − 1.07·9-s + (−0.825 − 4.74i)10-s + (−2.63 − 2.63i)11-s + (0.936 + 2.61i)12-s + (1.93 + 3.04i)13-s + (−0.250 + 0.0435i)14-s + (−3.34 − 3.34i)15-s + (−3.08 + 2.54i)16-s + 4.33i·17-s + ⋯
L(s)  = 1  + (0.817 + 0.575i)2-s + 0.801·3-s + (0.337 + 0.941i)4-s + (−1.07 − 1.07i)5-s + (0.655 + 0.461i)6-s + (−0.0480 + 0.0480i)7-s + (−0.265 + 0.964i)8-s − 0.357·9-s + (−0.260 − 1.50i)10-s + (−0.793 − 0.793i)11-s + (0.270 + 0.754i)12-s + (0.537 + 0.843i)13-s + (−0.0668 + 0.0116i)14-s + (−0.863 − 0.863i)15-s + (−0.772 + 0.635i)16-s + 1.05i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.829 - 0.558i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :1/2),\ 0.829 - 0.558i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50434 + 0.459233i\)
\(L(\frac12)\) \(\approx\) \(1.50434 + 0.459233i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 - 0.813i)T \)
13 \( 1 + (-1.93 - 3.04i)T \)
good3 \( 1 - 1.38T + 3T^{2} \)
5 \( 1 + (2.40 + 2.40i)T + 5iT^{2} \)
7 \( 1 + (0.127 - 0.127i)T - 7iT^{2} \)
11 \( 1 + (2.63 + 2.63i)T + 11iT^{2} \)
17 \( 1 - 4.33iT - 17T^{2} \)
19 \( 1 + (-4.97 + 4.97i)T - 19iT^{2} \)
23 \( 1 - 3.98T + 23T^{2} \)
29 \( 1 + 4.59iT - 29T^{2} \)
31 \( 1 + (-1.07 - 1.07i)T + 31iT^{2} \)
37 \( 1 + (2.45 - 2.45i)T - 37iT^{2} \)
41 \( 1 + (-0.388 + 0.388i)T - 41iT^{2} \)
43 \( 1 + 5.02iT - 43T^{2} \)
47 \( 1 + (-1.00 + 1.00i)T - 47iT^{2} \)
53 \( 1 + 1.83iT - 53T^{2} \)
59 \( 1 + (-6.28 - 6.28i)T + 59iT^{2} \)
61 \( 1 - 10.5iT - 61T^{2} \)
67 \( 1 + (4.21 - 4.21i)T - 67iT^{2} \)
71 \( 1 + (2.59 + 2.59i)T + 71iT^{2} \)
73 \( 1 + (0.388 + 0.388i)T + 73iT^{2} \)
79 \( 1 + 15.3iT - 79T^{2} \)
83 \( 1 + (2.46 - 2.46i)T - 83iT^{2} \)
89 \( 1 + (11.3 + 11.3i)T + 89iT^{2} \)
97 \( 1 + (-3.31 + 3.31i)T - 97iT^{2} \)
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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

−13.71331418011072538241207872032, −13.21933553830647714063162284847, −11.97733926843294803718997268531, −11.19368962892920551559682897028, −8.869668120273623231966486408851, −8.432287381874690846597059988580, −7.37571114040786143428908254357, −5.65347815859101825072903709678, −4.33948708867299266456859597164, −3.11792598352996510764840247593, 2.81960985573929096596803077258, 3.56044134354854554941063967510, 5.31427496746830413111692945808, 7.05762134595474572748386341933, 7.981637153956632485227531968440, 9.677074953147536704181633494198, 10.76525484983058202798160627459, 11.59238934718402845552404258499, 12.68454633718648644487177692331, 13.84102658207858165518266712765

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