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Properties

Label 2-102-51.47-c4-0-16
Degree $2$
Conductor $102$
Sign $-0.329 + 0.944i$
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 + (−1.72 − 8.83i)3-s + 8.00·4-s + (22.2 − 22.2i)5-s + (4.88 + 24.9i)6-s + (54.5 − 54.5i)7-s − 22.6·8-s + (−75.0 + 30.5i)9-s + (−62.8 + 62.8i)10-s + (109. + 109. i)11-s + (−13.8 − 70.6i)12-s + 294.·13-s + (−154. + 154. i)14-s + (−234. − 157. i)15-s + 64.0·16-s + (−286. + 38.5i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.192 − 0.981i)3-s + 0.500·4-s + (0.888 − 0.888i)5-s + (0.135 + 0.693i)6-s + (1.11 − 1.11i)7-s − 0.353·8-s + (−0.926 + 0.377i)9-s + (−0.628 + 0.628i)10-s + (0.908 + 0.908i)11-s + (−0.0960 − 0.490i)12-s + 1.74·13-s + (−0.786 + 0.786i)14-s + (−1.04 − 0.701i)15-s + 0.250·16-s + (−0.991 + 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.329 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $-0.329 + 0.944i$
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: \((2,\ 102,\ (\ :2),\ -0.329 + 0.944i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.902454 - 1.27084i\)
\(L(\frac12)\) \(\approx\) \(0.902454 - 1.27084i\)
\(L(3)\) 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 \( 1 + (1.72 + 8.83i)T \)
17 \( 1 + (286. - 38.5i)T \)
good5 \( 1 + (-22.2 + 22.2i)T - 625iT^{2} \)
7 \( 1 + (-54.5 + 54.5i)T - 2.40e3iT^{2} \)
11 \( 1 + (-109. - 109. i)T + 1.46e4iT^{2} \)
13 \( 1 - 294.T + 2.85e4T^{2} \)
19 \( 1 + 502. iT - 1.30e5T^{2} \)
23 \( 1 + (133. + 133. i)T + 2.79e5iT^{2} \)
29 \( 1 + (234. - 234. i)T - 7.07e5iT^{2} \)
31 \( 1 + (509. + 509. i)T + 9.23e5iT^{2} \)
37 \( 1 + (-204. - 204. i)T + 1.87e6iT^{2} \)
41 \( 1 + (786. + 786. i)T + 2.82e6iT^{2} \)
43 \( 1 - 3.25e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.51e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.73e3T + 7.89e6T^{2} \)
59 \( 1 - 3.52e3T + 1.21e7T^{2} \)
61 \( 1 + (2.07e3 - 2.07e3i)T - 1.38e7iT^{2} \)
67 \( 1 - 2.05e3T + 2.01e7T^{2} \)
71 \( 1 + (5.26e3 - 5.26e3i)T - 2.54e7iT^{2} \)
73 \( 1 + (5.77e3 + 5.77e3i)T + 2.83e7iT^{2} \)
79 \( 1 + (964. - 964. i)T - 3.89e7iT^{2} \)
83 \( 1 - 5.52e3T + 4.74e7T^{2} \)
89 \( 1 - 23.1iT - 6.27e7T^{2} \)
97 \( 1 + (-3.13e3 - 3.13e3i)T + 8.85e7iT^{2} \)
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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

−12.99951345242578051860289313644, −11.49018069516352176978216936009, −10.85942132053692717103027249087, −9.246700408850371011540342339884, −8.438638483591842897957769105929, −7.21134506882404050833338182413, −6.17895308422227682306467661748, −4.56124235420069855720452023147, −1.76002743494580942119325957279, −1.04488733338755955794618755452, 1.86045083698695369597804306633, 3.57935745091243583753893181736, 5.66195061233006511823882797946, 6.30112450657933636061222209602, 8.476451475160369873565866324710, 8.972335253287888136096658821610, 10.29760166220063317313544127616, 11.13671334988666671237346674827, 11.73785090283077140132214540371, 13.82905484988846973864725253612

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