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Properties

Label 2-195-13.4-c1-0-4
Degree $2$
Conductor $195$
Sign $0.172 - 0.985i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
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  + (2.10 + 1.21i)2-s + (−0.5 + 0.866i)3-s + (1.95 + 3.38i)4-s − i·5-s + (−2.10 + 1.21i)6-s + (1.12 − 0.650i)7-s + 4.64i·8-s + (−0.499 − 0.866i)9-s + (1.21 − 2.10i)10-s + (−2.75 − 1.58i)11-s − 3.90·12-s + (−3.40 + 1.19i)13-s + 3.16·14-s + (0.866 + 0.5i)15-s + (−1.73 + 3.00i)16-s + (0.325 + 0.564i)17-s + ⋯
L(s)  = 1  + (1.48 + 0.859i)2-s + (−0.288 + 0.499i)3-s + (0.977 + 1.69i)4-s − 0.447i·5-s + (−0.859 + 0.496i)6-s + (0.425 − 0.245i)7-s + 1.64i·8-s + (−0.166 − 0.288i)9-s + (0.384 − 0.665i)10-s + (−0.829 − 0.478i)11-s − 1.12·12-s + (−0.943 + 0.331i)13-s + 0.845·14-s + (0.223 + 0.129i)15-s + (−0.433 + 0.750i)16-s + (0.0789 + 0.136i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.172 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.172 - 0.985i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.172 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72490 + 1.44919i\)
\(L(\frac12)\) \(\approx\) \(1.72490 + 1.44919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + iT \)
13 \( 1 + (3.40 - 1.19i)T \)
good2 \( 1 + (-2.10 - 1.21i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-1.12 + 0.650i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.75 + 1.58i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.325 - 0.564i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.17 + 1.25i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.889 + 1.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.18 + 7.25i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.50iT - 31T^{2} \)
37 \( 1 + (-1.21 - 0.698i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.0 + 5.81i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.63 - 2.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.2iT - 47T^{2} \)
53 \( 1 + 6.35T + 53T^{2} \)
59 \( 1 + (5.18 - 2.99i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.49 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.3 - 6.57i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.949 - 0.548i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.2iT - 73T^{2} \)
79 \( 1 + 1.33T + 79T^{2} \)
83 \( 1 + 14.2iT - 83T^{2} \)
89 \( 1 + (8.31 + 4.80i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.88 + 3.97i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.82666421185265772941062312609, −12.09104758951871993514400414607, −11.10195133502228345094414159227, −9.848502243129502341593203033421, −8.354274943666394142853264763732, −7.36172842089177478196729990131, −6.16035277531647695585474630627, −5.05050086022286102951404540229, −4.53328819988093850266447702136, −3.02824397080268767756018759427, 2.03204988282098085916443255775, 3.16463908132796041242398484229, 4.82632783327145382806458308330, 5.50539819132936184318055797532, 6.83013998463574117066669186054, 7.953592159533884023308570392905, 9.878349198268338175967990163686, 10.71534706150888715571520221900, 11.67432357139829375004850280660, 12.28275911717957593886308496105

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