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Properties

Label 2-195-65.49-c1-0-15
Degree $2$
Conductor $195$
Sign $-0.699 - 0.714i$
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  + (−1.00 − 1.74i)2-s + (−0.866 + 0.5i)3-s + (−1.02 + 1.77i)4-s + (−1.28 − 1.83i)5-s + (1.74 + 1.00i)6-s + (1.11 − 1.92i)7-s + 0.0905·8-s + (0.499 − 0.866i)9-s + (−1.89 + 4.07i)10-s + (−4.40 + 2.54i)11-s − 2.04i·12-s + (−1.22 + 3.38i)13-s − 4.47·14-s + (2.02 + 0.944i)15-s + (1.95 + 3.38i)16-s + (−2.00 − 1.15i)17-s + ⋯
L(s)  = 1  + (−0.711 − 1.23i)2-s + (−0.499 + 0.288i)3-s + (−0.511 + 0.885i)4-s + (−0.573 − 0.818i)5-s + (0.711 + 0.410i)6-s + (0.420 − 0.727i)7-s + 0.0320·8-s + (0.166 − 0.288i)9-s + (−0.600 + 1.28i)10-s + (−1.32 + 0.767i)11-s − 0.590i·12-s + (−0.340 + 0.940i)13-s − 1.19·14-s + (0.523 + 0.243i)15-s + (0.488 + 0.846i)16-s + (−0.485 − 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.699 - 0.714i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.699 - 0.714i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.101048 + 0.240247i\)
\(L(\frac12)\) \(\approx\) \(0.101048 + 0.240247i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.28 + 1.83i)T \)
13 \( 1 + (1.22 - 3.38i)T \)
good2 \( 1 + (1.00 + 1.74i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-1.11 + 1.92i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.40 - 2.54i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.00 + 1.15i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.07 + 3.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.77 + 2.75i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.503 - 0.872i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.55iT - 31T^{2} \)
37 \( 1 + (0.579 + 1.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.42 + 0.823i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.97 + 4.60i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.34T + 47T^{2} \)
53 \( 1 + 9.36iT - 53T^{2} \)
59 \( 1 + (-0.894 - 0.516i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.32 + 9.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 - 2.88i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.54 - 0.889i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.49T + 73T^{2} \)
79 \( 1 + 7.05T + 79T^{2} \)
83 \( 1 + 2.25T + 83T^{2} \)
89 \( 1 + (7.03 - 4.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.01 + 5.22i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.53758486817027219895562826758, −11.01223068364254515007841897784, −10.11525639137275213967769560962, −9.157996431557444804183015519132, −8.186434284963673636077593861099, −6.92796539727657912980196407665, −4.96119384471705354928780382882, −4.15923126629921145600335990360, −2.19709643970459425959610131307, −0.28966449059066452117023065355, 2.90871185564924216366509163050, 5.16824452408839105293540270471, 6.05865277376457654838438836148, 7.07657701694446225271637330247, 8.077881855721780340904761683374, 8.542949604130213240525397664238, 10.25996443971873476724386197144, 10.95001270568787430737342066272, 12.10411121422429177974125359020, 13.09185344745430189735396251481

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