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Properties

Label 2-195-65.29-c1-0-6
Degree $2$
Conductor $195$
Sign $0.792 - 0.610i$
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  + (−0.729 + 0.421i)2-s + (0.866 − 0.5i)3-s + (−0.644 + 1.11i)4-s + (2.23 − 0.0545i)5-s + (−0.421 + 0.729i)6-s + (0.347 + 0.200i)7-s − 2.77i·8-s + (0.499 − 0.866i)9-s + (−1.60 + 0.981i)10-s + (2.45 + 4.24i)11-s + 1.28i·12-s + (−3.55 − 0.572i)13-s − 0.338·14-s + (1.90 − 1.16i)15-s + (−0.121 − 0.211i)16-s + (6.13 + 3.54i)17-s + ⋯
L(s)  = 1  + (−0.516 + 0.297i)2-s + (0.499 − 0.288i)3-s + (−0.322 + 0.558i)4-s + (0.999 − 0.0244i)5-s + (−0.172 + 0.297i)6-s + (0.131 + 0.0758i)7-s − 0.980i·8-s + (0.166 − 0.288i)9-s + (−0.508 + 0.310i)10-s + (0.739 + 1.28i)11-s + 0.372i·12-s + (−0.987 − 0.158i)13-s − 0.0903·14-s + (0.492 − 0.300i)15-s + (−0.0304 − 0.0528i)16-s + (1.48 + 0.858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.792 - 0.610i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.792 - 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09834 + 0.374146i\)
\(L(\frac12)\) \(\approx\) \(1.09834 + 0.374146i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-2.23 + 0.0545i)T \)
13 \( 1 + (3.55 + 0.572i)T \)
good2 \( 1 + (0.729 - 0.421i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-0.347 - 0.200i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.45 - 4.24i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-6.13 - 3.54i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.12 + 1.95i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.861 - 0.497i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.94 + 6.82i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.30T + 31T^{2} \)
37 \( 1 + (7.62 - 4.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.65 + 4.60i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.74 + 1.00i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.62iT - 47T^{2} \)
53 \( 1 + 10.8iT - 53T^{2} \)
59 \( 1 + (1.52 - 2.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.55 + 6.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.32 - 3.07i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.16 - 5.48i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.01iT - 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 - 10.5iT - 83T^{2} \)
89 \( 1 + (-0.262 - 0.454i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.15 + 4.71i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.62718170129265330984562732495, −11.99799684843141971615926023862, −9.981345047964544052101062700522, −9.707716701993908903541875405794, −8.626562684173553337671007956957, −7.55147188685797692661391932641, −6.77025582730832670644085645215, −5.20686998284855043147097561209, −3.65585670648722972741573940728, −1.92136138185929465858684980216, 1.51236730318976398783653811484, 3.17537425834857660085146808757, 5.05427129879316551474207113491, 5.89475384049561041256749274495, 7.50475854604963369282572949408, 8.874939170867064790187809301930, 9.402323938312417953354709583258, 10.22660904291063271407300253265, 11.10476324664074627502468176811, 12.34419904315835181694911096567

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