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Properties

Label 2-195-195.173-c1-0-9
Degree $2$
Conductor $195$
Sign $0.996 + 0.0803i$
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  + (−0.283 − 1.05i)2-s + (0.872 + 1.49i)3-s + (0.689 − 0.397i)4-s + (1.65 + 1.50i)5-s + (1.33 − 1.34i)6-s + (−0.0430 + 0.160i)7-s + (−2.16 − 2.16i)8-s + (−1.47 + 2.61i)9-s + (1.12 − 2.17i)10-s + (−2.43 + 4.22i)11-s + (1.19 + 0.684i)12-s + (2.82 − 2.24i)13-s + 0.182·14-s + (−0.813 + 3.78i)15-s + (−0.887 + 1.53i)16-s + (1.24 − 4.65i)17-s + ⋯
L(s)  = 1  + (−0.200 − 0.749i)2-s + (0.503 + 0.863i)3-s + (0.344 − 0.198i)4-s + (0.738 + 0.674i)5-s + (0.546 − 0.551i)6-s + (−0.0162 + 0.0606i)7-s + (−0.766 − 0.766i)8-s + (−0.492 + 0.870i)9-s + (0.356 − 0.688i)10-s + (−0.734 + 1.27i)11-s + (0.345 + 0.197i)12-s + (0.782 − 0.622i)13-s + 0.0487·14-s + (−0.210 + 0.977i)15-s + (−0.221 + 0.384i)16-s + (0.302 − 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.996 + 0.0803i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.996 + 0.0803i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45243 - 0.0584749i\)
\(L(\frac12)\) \(\approx\) \(1.45243 - 0.0584749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.872 - 1.49i)T \)
5 \( 1 + (-1.65 - 1.50i)T \)
13 \( 1 + (-2.82 + 2.24i)T \)
good2 \( 1 + (0.283 + 1.05i)T + (-1.73 + i)T^{2} \)
7 \( 1 + (0.0430 - 0.160i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.43 - 4.22i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.24 + 4.65i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.37 + 2.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.51 + 5.64i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.47 + 4.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.41iT - 31T^{2} \)
37 \( 1 + (4.48 - 1.20i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.16 - 3.74i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.340 - 1.27i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.58 + 3.58i)T - 47iT^{2} \)
53 \( 1 + (3.34 - 3.34i)T - 53iT^{2} \)
59 \( 1 + (8.49 - 4.90i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.20 + 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.702 - 0.188i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.37 + 2.38i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.45 - 5.45i)T - 73iT^{2} \)
79 \( 1 - 7.39iT - 79T^{2} \)
83 \( 1 + (1.60 + 1.60i)T + 83iT^{2} \)
89 \( 1 + (-5.98 - 3.45i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.36 + 16.2i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.36763002248882281996912052244, −11.13401161018875490035986175677, −10.35398251288212420613180392226, −9.941639285161036060369155453054, −8.922079292801527751055763697028, −7.43653566717210340868230472621, −6.16541417197240884533476370694, −4.82685672139763895145786464792, −3.09802830784046390491220063675, −2.26918789880476658397366233644, 1.76922169502104022419159865143, 3.39389197378025695125000420211, 5.74321725223739388083185182535, 6.18713339274416726897744213252, 7.52944981894079012279171674206, 8.472114247724945025799834498109, 8.922067192088285034165321199005, 10.55400749771768097974149472894, 11.76168116072437820182351363499, 12.68811729652318509111399026582

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