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Properties

Label 2-195-195.38-c1-0-12
Degree $2$
Conductor $195$
Sign $0.838 - 0.544i$
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  + (−0.299 + 0.299i)2-s + (1.72 − 0.125i)3-s + 1.82i·4-s + (2.19 − 0.414i)5-s + (−0.479 + 0.554i)6-s + (−0.976 − 0.976i)7-s + (−1.14 − 1.14i)8-s + (2.96 − 0.431i)9-s + (−0.533 + 0.781i)10-s − 1.78·11-s + (0.227 + 3.14i)12-s + (−2.32 + 2.75i)13-s + 0.584·14-s + (3.74 − 0.990i)15-s − 2.95·16-s + (−3.26 − 3.26i)17-s + ⋯
L(s)  = 1  + (−0.211 + 0.211i)2-s + (0.997 − 0.0721i)3-s + 0.910i·4-s + (0.982 − 0.185i)5-s + (−0.195 + 0.226i)6-s + (−0.368 − 0.368i)7-s + (−0.404 − 0.404i)8-s + (0.989 − 0.143i)9-s + (−0.168 + 0.247i)10-s − 0.537·11-s + (0.0657 + 0.908i)12-s + (−0.643 + 0.765i)13-s + 0.156·14-s + (0.966 − 0.255i)15-s − 0.739·16-s + (−0.790 − 0.790i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.838 - 0.544i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.838 - 0.544i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46149 + 0.433067i\)
\(L(\frac12)\) \(\approx\) \(1.46149 + 0.433067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.72 + 0.125i)T \)
5 \( 1 + (-2.19 + 0.414i)T \)
13 \( 1 + (2.32 - 2.75i)T \)
good2 \( 1 + (0.299 - 0.299i)T - 2iT^{2} \)
7 \( 1 + (0.976 + 0.976i)T + 7iT^{2} \)
11 \( 1 + 1.78T + 11T^{2} \)
17 \( 1 + (3.26 + 3.26i)T + 17iT^{2} \)
19 \( 1 - 6.18T + 19T^{2} \)
23 \( 1 + (-0.696 + 0.696i)T - 23iT^{2} \)
29 \( 1 + 7.33T + 29T^{2} \)
31 \( 1 - 6.61iT - 31T^{2} \)
37 \( 1 + (5.21 + 5.21i)T + 37iT^{2} \)
41 \( 1 + 6.45T + 41T^{2} \)
43 \( 1 + (1.78 + 1.78i)T + 43iT^{2} \)
47 \( 1 + (-5.69 + 5.69i)T - 47iT^{2} \)
53 \( 1 + (1.74 - 1.74i)T - 53iT^{2} \)
59 \( 1 - 9.15iT - 59T^{2} \)
61 \( 1 - 1.01T + 61T^{2} \)
67 \( 1 + (3.72 + 3.72i)T + 67iT^{2} \)
71 \( 1 - 5.49T + 71T^{2} \)
73 \( 1 + (-1.35 + 1.35i)T - 73iT^{2} \)
79 \( 1 - 10.2iT - 79T^{2} \)
83 \( 1 + (7.93 + 7.93i)T + 83iT^{2} \)
89 \( 1 - 3.75iT - 89T^{2} \)
97 \( 1 + (-6.92 - 6.92i)T + 97iT^{2} \)
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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

−12.87016110400973775409956265361, −11.85854139036652442577224447573, −10.27736559071254346198541973444, −9.319086034029838600968771970461, −8.806006855150748579189088713783, −7.39076067209975349985572779652, −6.89251171985681651512743157498, −5.00752147582323177160742657322, −3.49403919483253015186214309893, −2.28063219249753663380681192224, 1.88806099426389216575016440729, 3.01634738618464354722191785729, 5.04662363486820841760961006797, 6.04447783207447774822852441208, 7.37757356653980381474836818380, 8.719667920693660030075057479170, 9.669262449812916113101426655649, 10.05034749728898634491457705726, 11.14096763456124860118445577751, 12.73636791858483160666396089040

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