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Properties

Label 2-102-17.4-c3-0-0
Degree $2$
Conductor $102$
Sign $-0.901 + 0.431i$
Analytic cond. $6.01819$
Root an. cond. $2.45320$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + (2.12 − 2.12i)3-s − 4·4-s + (−11.8 + 11.8i)5-s + (4.24 + 4.24i)6-s + (−13.2 − 13.2i)7-s − 8i·8-s − 8.99i·9-s + (−23.6 − 23.6i)10-s + (0.845 + 0.845i)11-s + (−8.48 + 8.48i)12-s − 68.4·13-s + (26.4 − 26.4i)14-s + 50.2i·15-s + 16·16-s + (−59.0 + 37.7i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (−1.05 + 1.05i)5-s + (0.288 + 0.288i)6-s + (−0.714 − 0.714i)7-s − 0.353i·8-s − 0.333i·9-s + (−0.748 − 0.748i)10-s + (0.0231 + 0.0231i)11-s + (−0.204 + 0.204i)12-s − 1.45·13-s + (0.505 − 0.505i)14-s + 0.864i·15-s + 0.250·16-s + (−0.842 + 0.538i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $-0.901 + 0.431i$
Analytic conductor: \(6.01819\)
Root analytic conductor: \(2.45320\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{102} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :3/2),\ -0.901 + 0.431i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0459533 - 0.202412i\)
\(L(\frac12)\) \(\approx\) \(0.0459533 - 0.202412i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
3 \( 1 + (-2.12 + 2.12i)T \)
17 \( 1 + (59.0 - 37.7i)T \)
good5 \( 1 + (11.8 - 11.8i)T - 125iT^{2} \)
7 \( 1 + (13.2 + 13.2i)T + 343iT^{2} \)
11 \( 1 + (-0.845 - 0.845i)T + 1.33e3iT^{2} \)
13 \( 1 + 68.4T + 2.19e3T^{2} \)
19 \( 1 - 5.36iT - 6.85e3T^{2} \)
23 \( 1 + (-26.9 - 26.9i)T + 1.21e4iT^{2} \)
29 \( 1 + (-40.7 + 40.7i)T - 2.43e4iT^{2} \)
31 \( 1 + (-41.3 + 41.3i)T - 2.97e4iT^{2} \)
37 \( 1 + (226. - 226. i)T - 5.06e4iT^{2} \)
41 \( 1 + (-218. - 218. i)T + 6.89e4iT^{2} \)
43 \( 1 - 281. iT - 7.95e4T^{2} \)
47 \( 1 - 274.T + 1.03e5T^{2} \)
53 \( 1 - 392. iT - 1.48e5T^{2} \)
59 \( 1 + 764. iT - 2.05e5T^{2} \)
61 \( 1 + (358. + 358. i)T + 2.26e5iT^{2} \)
67 \( 1 + 782.T + 3.00e5T^{2} \)
71 \( 1 + (785. - 785. i)T - 3.57e5iT^{2} \)
73 \( 1 + (-496. + 496. i)T - 3.89e5iT^{2} \)
79 \( 1 + (-558. - 558. i)T + 4.93e5iT^{2} \)
83 \( 1 + 646. iT - 5.71e5T^{2} \)
89 \( 1 + 385.T + 7.04e5T^{2} \)
97 \( 1 + (-1.13e3 + 1.13e3i)T - 9.12e5iT^{2} \)
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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

−14.14241788697765065590239357322, −13.08265456214107249362636860575, −11.98565271012604189312161936755, −10.67264938184714874532582176871, −9.554845376132833811033454317301, −8.025557409517394733547834316908, −7.21891887476946247683274485608, −6.49921262866970629537026130425, −4.34522611785397315568977101330, −3.02411305432723487295327035562, 0.10650488536020613552800096927, 2.62876793208126819816941993873, 4.14770038987384928941504022562, 5.19077391305736792903975923942, 7.37552339071660238947900966667, 8.785392921345086868746231705217, 9.259849521428599028895054523129, 10.60122400920014282260432519599, 12.08055160623598986783980051358, 12.34302636494515056049948119641

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