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Properties

Label 2-195-39.5-c1-0-15
Degree $2$
Conductor $195$
Sign $0.388 + 0.921i$
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  + (0.789 + 0.789i)2-s + (−1.64 − 0.548i)3-s − 0.754i·4-s + (−0.707 − 0.707i)5-s + (−0.863 − 1.72i)6-s + (−1.97 − 1.97i)7-s + (2.17 − 2.17i)8-s + (2.39 + 1.80i)9-s − 1.11i·10-s + (3.56 − 3.56i)11-s + (−0.413 + 1.23i)12-s + (−3.37 − 1.26i)13-s − 3.12i·14-s + (0.773 + 1.54i)15-s + 1.92·16-s + 0.700·17-s + ⋯
L(s)  = 1  + (0.558 + 0.558i)2-s + (−0.948 − 0.316i)3-s − 0.377i·4-s + (−0.316 − 0.316i)5-s + (−0.352 − 0.706i)6-s + (−0.747 − 0.747i)7-s + (0.768 − 0.768i)8-s + (0.799 + 0.600i)9-s − 0.352i·10-s + (1.07 − 1.07i)11-s + (−0.119 + 0.357i)12-s + (−0.936 − 0.350i)13-s − 0.834i·14-s + (0.199 + 0.400i)15-s + 0.480·16-s + 0.169·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.853391 - 0.566094i\)
\(L(\frac12)\) \(\approx\) \(0.853391 - 0.566094i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.64 + 0.548i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (3.37 + 1.26i)T \)
good2 \( 1 + (-0.789 - 0.789i)T + 2iT^{2} \)
7 \( 1 + (1.97 + 1.97i)T + 7iT^{2} \)
11 \( 1 + (-3.56 + 3.56i)T - 11iT^{2} \)
17 \( 1 - 0.700T + 17T^{2} \)
19 \( 1 + (4.32 - 4.32i)T - 19iT^{2} \)
23 \( 1 - 4.02T + 23T^{2} \)
29 \( 1 - 6.96iT - 29T^{2} \)
31 \( 1 + (-6.53 + 6.53i)T - 31iT^{2} \)
37 \( 1 + (-2.65 - 2.65i)T + 37iT^{2} \)
41 \( 1 + (-5.37 - 5.37i)T + 41iT^{2} \)
43 \( 1 + 1.62iT - 43T^{2} \)
47 \( 1 + (3.82 - 3.82i)T - 47iT^{2} \)
53 \( 1 - 0.258iT - 53T^{2} \)
59 \( 1 + (-9.68 + 9.68i)T - 59iT^{2} \)
61 \( 1 + 3.98T + 61T^{2} \)
67 \( 1 + (-4.24 + 4.24i)T - 67iT^{2} \)
71 \( 1 + (2.54 + 2.54i)T + 71iT^{2} \)
73 \( 1 + (-6.24 - 6.24i)T + 73iT^{2} \)
79 \( 1 - 0.921T + 79T^{2} \)
83 \( 1 + (-8.89 - 8.89i)T + 83iT^{2} \)
89 \( 1 + (-9.13 + 9.13i)T - 89iT^{2} \)
97 \( 1 + (1.54 - 1.54i)T - 97iT^{2} \)
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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

−12.56930336991376345999841539288, −11.40113614567653137453166096559, −10.46840371903795789152340735771, −9.580051233164708629484963109692, −7.898010185863162078554787557222, −6.72790349990886706174251942479, −6.17067213473763523225059457507, −4.96426588672423676959185976642, −3.86022514924349542464743906873, −0.905180716470870183288817385259, 2.51590850925477213582120728324, 4.05758054034511172109581231374, 4.88077480129934605241072752813, 6.44911075716865198123025145462, 7.24798108455813484250182195338, 8.977290981532882365160511045950, 9.931194091346365776985642806860, 11.05600784804621108820076937038, 12.02546449431804751807925355294, 12.26459577512155497171146436703

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