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Properties

Label 2-195-39.5-c1-0-13
Degree $2$
Conductor $195$
Sign $-0.428 + 0.903i$
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  + (−0.455 − 0.455i)2-s + (−0.446 + 1.67i)3-s − 1.58i·4-s + (−0.707 − 0.707i)5-s + (0.966 − 0.559i)6-s + (−2.89 − 2.89i)7-s + (−1.63 + 1.63i)8-s + (−2.60 − 1.49i)9-s + 0.644i·10-s + (2.45 − 2.45i)11-s + (2.65 + 0.707i)12-s + (3.43 − 1.09i)13-s + 2.64i·14-s + (1.49 − 0.867i)15-s − 1.68·16-s − 6.48·17-s + ⋯
L(s)  = 1  + (−0.322 − 0.322i)2-s + (−0.257 + 0.966i)3-s − 0.792i·4-s + (−0.316 − 0.316i)5-s + (0.394 − 0.228i)6-s + (−1.09 − 1.09i)7-s + (−0.577 + 0.577i)8-s + (−0.867 − 0.498i)9-s + 0.203i·10-s + (0.741 − 0.741i)11-s + (0.765 + 0.204i)12-s + (0.952 − 0.304i)13-s + 0.705i·14-s + (0.387 − 0.224i)15-s − 0.420·16-s − 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.428 + 0.903i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.428 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.428 + 0.903i$
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: \((2,\ 195,\ (\ :1/2),\ -0.428 + 0.903i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.318312 - 0.503116i\)
\(L(\frac12)\) \(\approx\) \(0.318312 - 0.503116i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.446 - 1.67i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-3.43 + 1.09i)T \)
good2 \( 1 + (0.455 + 0.455i)T + 2iT^{2} \)
7 \( 1 + (2.89 + 2.89i)T + 7iT^{2} \)
11 \( 1 + (-2.45 + 2.45i)T - 11iT^{2} \)
17 \( 1 + 6.48T + 17T^{2} \)
19 \( 1 + (-3.48 + 3.48i)T - 19iT^{2} \)
23 \( 1 + 2.39T + 23T^{2} \)
29 \( 1 - 0.0728iT - 29T^{2} \)
31 \( 1 + (3.16 - 3.16i)T - 31iT^{2} \)
37 \( 1 + (-6.34 - 6.34i)T + 37iT^{2} \)
41 \( 1 + (2.12 + 2.12i)T + 41iT^{2} \)
43 \( 1 + 3.53iT - 43T^{2} \)
47 \( 1 + (-8.05 + 8.05i)T - 47iT^{2} \)
53 \( 1 - 9.32iT - 53T^{2} \)
59 \( 1 + (-3.31 + 3.31i)T - 59iT^{2} \)
61 \( 1 + 1.29T + 61T^{2} \)
67 \( 1 + (-0.922 + 0.922i)T - 67iT^{2} \)
71 \( 1 + (2.96 + 2.96i)T + 71iT^{2} \)
73 \( 1 + (5.91 + 5.91i)T + 73iT^{2} \)
79 \( 1 - 9.54T + 79T^{2} \)
83 \( 1 + (2.62 + 2.62i)T + 83iT^{2} \)
89 \( 1 + (-7.03 + 7.03i)T - 89iT^{2} \)
97 \( 1 + (-6.77 + 6.77i)T - 97iT^{2} \)
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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

−11.75045548898178841033441582872, −10.97698486677892819940272894868, −10.34500357513156211544435781717, −9.294782311169195148662207044934, −8.730949843915243286524031486571, −6.76256782854025604787556421992, −5.86853379004536206023222216156, −4.43033297857164547222865215476, −3.34443779740084924215355080472, −0.59215546641218459801928964821, 2.42828361024714422753983145847, 3.85597679652573631100072190809, 6.05263741192474777071687152761, 6.64453829705425384946422898835, 7.64378103159064947254890507604, 8.753236862218706794738860297785, 9.457063082899106645258822712921, 11.26668177999442739912115902934, 11.98082903800582551002211119894, 12.71681345616399609008955273913

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