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Properties

Label 2-195-65.49-c1-0-10
Degree $2$
Conductor $195$
Sign $0.433 - 0.901i$
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 + (4.47 + 0.393i)10-s + (−4.40 + 2.54i)11-s + 2.04i·12-s + (1.22 − 3.38i)13-s − 4.47·14-s + (0.195 − 2.22i)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 + (1.41 + 0.124i)10-s + (−1.32 + 0.767i)11-s + 0.590i·12-s + (0.340 − 0.940i)13-s − 1.19·14-s + (0.0505 − 0.575i)15-s + (0.488 + 0.846i)16-s + (0.485 + 0.280i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.433 - 0.901i$
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: \((2,\ 195,\ (\ :1/2),\ 0.433 - 0.901i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64690 + 1.03566i\)
\(L(\frac12)\) \(\approx\) \(1.64690 + 1.03566i\)
\(L(\frac{3}{2})\) 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

−12.95045914483003554096789770739, −12.47197602409513446121306426843, −10.53638999092796280647709252888, −9.453459218809863654171890854974, −8.269159481441469459614498203137, −7.64991379257234481080535490088, −6.15757373619547678983279772419, −5.54246352174206886684978120950, −4.34668689514578456511863571004, −2.38426236647866121480191841876, 2.14971155147749082983279931055, 3.24210336548136276797341692246, 4.21883412247453692251310267204, 5.75747941840560884043438163053, 7.13896928931004458285778552331, 8.477942254479492854471814253182, 10.05433427007758899858959586077, 10.39453058235763708620759883823, 11.16074412264008458474429270183, 12.46691463350141429933846873352

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