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Properties

Label 2-195-65.29-c1-0-2
Degree $2$
Conductor $195$
Sign $0.560 - 0.828i$
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  + (−1.52 + 0.881i)2-s + (−0.866 + 0.5i)3-s + (0.553 − 0.959i)4-s + (1.52 − 1.63i)5-s + (0.881 − 1.52i)6-s + (1.92 + 1.11i)7-s − 1.57i·8-s + (0.499 − 0.866i)9-s + (−0.889 + 3.84i)10-s + (−0.646 − 1.11i)11-s + 1.10i·12-s + (2.90 + 2.14i)13-s − 3.92·14-s + (−0.504 + 2.17i)15-s + (2.49 + 4.32i)16-s + (5.27 + 3.04i)17-s + ⋯
L(s)  = 1  + (−1.07 + 0.623i)2-s + (−0.499 + 0.288i)3-s + (0.276 − 0.479i)4-s + (0.682 − 0.730i)5-s + (0.359 − 0.623i)6-s + (0.728 + 0.420i)7-s − 0.556i·8-s + (0.166 − 0.288i)9-s + (−0.281 + 1.21i)10-s + (−0.194 − 0.337i)11-s + 0.319i·12-s + (0.804 + 0.593i)13-s − 1.04·14-s + (−0.130 + 0.562i)15-s + (0.623 + 1.08i)16-s + (1.27 + 0.738i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.560 - 0.828i$
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: \((2,\ 195,\ (\ :1/2),\ 0.560 - 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.626085 + 0.332217i\)
\(L(\frac12)\) \(\approx\) \(0.626085 + 0.332217i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 + 1.63i)T \)
13 \( 1 + (-2.90 - 2.14i)T \)
good2 \( 1 + (1.52 - 0.881i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (-1.92 - 1.11i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.646 + 1.11i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-5.27 - 3.04i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.01 - 3.49i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.29 - 1.90i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.38 + 2.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.4T + 31T^{2} \)
37 \( 1 + (-6.77 + 3.91i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.01 - 3.49i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (9.27 + 5.35i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.00iT - 47T^{2} \)
53 \( 1 - 2.34iT - 53T^{2} \)
59 \( 1 + (-5.29 + 9.17i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.84 - 4.93i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.84 - 1.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.02 - 5.23i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.10iT - 73T^{2} \)
79 \( 1 + 8.02T + 79T^{2} \)
83 \( 1 - 10.9iT - 83T^{2} \)
89 \( 1 + (0.387 + 0.670i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.94 + 4.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.53519767267359912725825851066, −11.59536768509386250476941199037, −10.28662182368813202923256491229, −9.654240818011706809516030210908, −8.470155414679327811619833535239, −8.051750437539948626025398787312, −6.32343541118274061681786342800, −5.62540522894253904228953605894, −4.10715310928544167539890598214, −1.40688086561026914557297832029, 1.23458754439027182049312997797, 2.74710718440517124458026236590, 4.96191570957772255817624299244, 6.17864245286595469274180127849, 7.49909261615548025323147299053, 8.365514433829121816620238062486, 9.766401479661464019879323141140, 10.36142382487938175731656829002, 11.13470143971291731343791755340, 11.89876297230137793115447563808

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