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Properties

Label 2-102-51.47-c4-0-20
Degree $2$
Conductor $102$
Sign $-0.952 + 0.305i$
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 + (6.09 − 6.62i)3-s + 8.00·4-s + (−17.8 + 17.8i)5-s + (−17.2 + 18.7i)6-s + (8.63 − 8.63i)7-s − 22.6·8-s + (−6.82 − 80.7i)9-s + (50.4 − 50.4i)10-s + (−112. − 112. i)11-s + (48.7 − 53.0i)12-s + 45.0·13-s + (−24.4 + 24.4i)14-s + (9.56 + 226. i)15-s + 64.0·16-s + (−285. + 42.0i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.676 − 0.736i)3-s + 0.500·4-s + (−0.713 + 0.713i)5-s + (−0.478 + 0.520i)6-s + (0.176 − 0.176i)7-s − 0.353·8-s + (−0.0842 − 0.996i)9-s + (0.504 − 0.504i)10-s + (−0.932 − 0.932i)11-s + (0.338 − 0.368i)12-s + 0.266·13-s + (−0.124 + 0.124i)14-s + (0.0425 + 1.00i)15-s + 0.250·16-s + (−0.989 + 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.952 + 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $-0.952 + 0.305i$
Analytic conductor: \(10.5437\)
Root analytic conductor: \(3.24711\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{102} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :2),\ -0.952 + 0.305i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0830031 - 0.529951i\)
\(L(\frac12)\) \(\approx\) \(0.0830031 - 0.529951i\)
\(L(3)\) 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 \( 1 + (-6.09 + 6.62i)T \)
17 \( 1 + (285. - 42.0i)T \)
good5 \( 1 + (17.8 - 17.8i)T - 625iT^{2} \)
7 \( 1 + (-8.63 + 8.63i)T - 2.40e3iT^{2} \)
11 \( 1 + (112. + 112. i)T + 1.46e4iT^{2} \)
13 \( 1 - 45.0T + 2.85e4T^{2} \)
19 \( 1 - 14.5iT - 1.30e5T^{2} \)
23 \( 1 + (550. + 550. i)T + 2.79e5iT^{2} \)
29 \( 1 + (494. - 494. i)T - 7.07e5iT^{2} \)
31 \( 1 + (1.06e3 + 1.06e3i)T + 9.23e5iT^{2} \)
37 \( 1 + (719. + 719. i)T + 1.87e6iT^{2} \)
41 \( 1 + (-1.54e3 - 1.54e3i)T + 2.82e6iT^{2} \)
43 \( 1 + 882. iT - 3.41e6T^{2} \)
47 \( 1 + 742. iT - 4.87e6T^{2} \)
53 \( 1 - 4.49e3T + 7.89e6T^{2} \)
59 \( 1 + 3.49e3T + 1.21e7T^{2} \)
61 \( 1 + (-1.35e3 + 1.35e3i)T - 1.38e7iT^{2} \)
67 \( 1 - 161.T + 2.01e7T^{2} \)
71 \( 1 + (-6.21e3 + 6.21e3i)T - 2.54e7iT^{2} \)
73 \( 1 + (680. + 680. i)T + 2.83e7iT^{2} \)
79 \( 1 + (-2.57e3 + 2.57e3i)T - 3.89e7iT^{2} \)
83 \( 1 - 6.64e3T + 4.74e7T^{2} \)
89 \( 1 - 809. iT - 6.27e7T^{2} \)
97 \( 1 + (75.7 + 75.7i)T + 8.85e7iT^{2} \)
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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

−12.64679570951961287711058865518, −11.33676280341285890086816481020, −10.64533687464388034397354280005, −9.062233822365865629592074941653, −8.081042586214801048052361551262, −7.34463598213376956795746284215, −6.13067386240474646075447577541, −3.65319271766176849238110228164, −2.29305162239886807372755822628, −0.26388525792944156211798917339, 2.14994526652032046963297903790, 3.96598775065690446702748785176, 5.23713677025052805020286507401, 7.38569938035671063657781516927, 8.267608470229215195272966368523, 9.146085426228163180014691167430, 10.18463727383309222294479891253, 11.24487915080018495624434572116, 12.43387106831122373663118279043, 13.59365545125943453693553323666

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