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Properties

Label 2-195-13.4-c1-0-2
Degree $2$
Conductor $195$
Sign $0.00919 + 0.999i$
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  + (−1.88 − 1.08i)2-s + (0.5 − 0.866i)3-s + (1.36 + 2.36i)4-s + i·5-s + (−1.88 + 1.08i)6-s + (3.38 − 1.95i)7-s − 1.59i·8-s + (−0.499 − 0.866i)9-s + (1.08 − 1.88i)10-s + (1.37 + 0.796i)11-s + 2.73·12-s + (3.12 − 1.79i)13-s − 8.49·14-s + (0.866 + 0.5i)15-s + (0.999 − 1.73i)16-s + (−1.88 − 3.26i)17-s + ⋯
L(s)  = 1  + (−1.33 − 0.769i)2-s + (0.288 − 0.499i)3-s + (0.683 + 1.18i)4-s + 0.447i·5-s + (−0.769 + 0.444i)6-s + (1.27 − 0.738i)7-s − 0.563i·8-s + (−0.166 − 0.288i)9-s + (0.343 − 0.595i)10-s + (0.415 + 0.240i)11-s + 0.788·12-s + (0.867 − 0.496i)13-s − 2.27·14-s + (0.223 + 0.129i)15-s + (0.249 − 0.433i)16-s + (−0.456 − 0.791i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00919 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00919 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.00919 + 0.999i$
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: \((2,\ 195,\ (\ :1/2),\ 0.00919 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.544965 - 0.539976i\)
\(L(\frac12)\) \(\approx\) \(0.544965 - 0.539976i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.12 + 1.79i)T \)
good2 \( 1 + (1.88 + 1.08i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-3.38 + 1.95i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.37 - 0.796i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.88 + 3.26i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.85 - 3.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.11 + 5.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.51 + 4.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.184iT - 31T^{2} \)
37 \( 1 + (1.42 + 0.824i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.98 - 5.18i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.37 - 5.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.58iT - 47T^{2} \)
53 \( 1 + 5.51T + 53T^{2} \)
59 \( 1 + (3.74 - 2.16i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.86 - 8.42i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.33 + 3.08i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.07 + 0.621i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 2.25iT - 73T^{2} \)
79 \( 1 + 4.33T + 79T^{2} \)
83 \( 1 + 9.12iT - 83T^{2} \)
89 \( 1 + (-7.92 - 4.57i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.29 - 4.79i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.88851921498681643969394812063, −10.92931065227168062705881305819, −10.57629859266954763289678318482, −9.226121754274832923496927829305, −8.249597486508668583661501583577, −7.68317715838936884925917624338, −6.39714795772663529550633192873, −4.33603822097459356006791073597, −2.54941345579326466271409966546, −1.20826748023369389669439955929, 1.72784026332288991859924928231, 4.18798187171938652736736844928, 5.57227582915398451291055662245, 6.79557984771137889350152393419, 8.184068617685733748263651156307, 8.744895363513499137064583488673, 9.206096585056966854733046324668, 10.72400613322895441392146676786, 11.26188009319816732380651605766, 12.71970144962076209876685100878

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