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Properties

Label 2-195-65.49-c1-0-4
Degree $2$
Conductor $195$
Sign $-0.682 - 0.730i$
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  + (1.30 + 2.26i)2-s + (0.866 − 0.5i)3-s + (−2.40 + 4.16i)4-s + (−2.09 + 0.794i)5-s + (2.26 + 1.30i)6-s + (−0.372 + 0.645i)7-s − 7.34·8-s + (0.499 − 0.866i)9-s + (−4.52 − 3.68i)10-s + (4.61 − 2.66i)11-s + 4.81i·12-s + (3.57 + 0.440i)13-s − 1.94·14-s + (−1.41 + 1.73i)15-s + (−4.77 − 8.26i)16-s + (−1.74 − 1.00i)17-s + ⋯
L(s)  = 1  + (0.922 + 1.59i)2-s + (0.499 − 0.288i)3-s + (−1.20 + 2.08i)4-s + (−0.934 + 0.355i)5-s + (0.922 + 0.532i)6-s + (−0.140 + 0.243i)7-s − 2.59·8-s + (0.166 − 0.288i)9-s + (−1.43 − 1.16i)10-s + (1.39 − 0.804i)11-s + 1.38i·12-s + (0.992 + 0.122i)13-s − 0.519·14-s + (−0.364 + 0.447i)15-s + (−1.19 − 2.06i)16-s + (−0.422 − 0.243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.682 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.682 - 0.730i$
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: \((2,\ 195,\ (\ :1/2),\ -0.682 - 0.730i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.715516 + 1.64760i\)
\(L(\frac12)\) \(\approx\) \(0.715516 + 1.64760i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 - 0.794i)T \)
13 \( 1 + (-3.57 - 0.440i)T \)
good2 \( 1 + (-1.30 - 2.26i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.372 - 0.645i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.61 + 2.66i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.74 + 1.00i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.92 - 2.26i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.28 - 2.47i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.47 + 7.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.43iT - 31T^{2} \)
37 \( 1 + (1.60 + 2.78i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.22 - 3.01i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.84 + 1.64i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.07T + 47T^{2} \)
53 \( 1 + 2.14iT - 53T^{2} \)
59 \( 1 + (-5.56 - 3.21i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.57 - 4.45i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.90 + 5.03i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.29 + 1.90i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 16.0T + 73T^{2} \)
79 \( 1 + 11.4T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + (2.30 - 1.33i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.570 + 0.987i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.40114188405353142792154570215, −12.06267450246598427401710184762, −11.47607432096305871489207277961, −9.322429603908236599029385421892, −8.381808626563732745742396343008, −7.65389688184853452721978994812, −6.59815357093094745827854384575, −5.84913376019999887965806855650, −4.06757482781687044353276318308, −3.49347174064973264322222936753, 1.50889602976134003042572024062, 3.40754293302611585040737577696, 4.01408893407294718268381802939, 5.05458149684337066897986072489, 6.83382420891763314575235526953, 8.590998727345815238259475356693, 9.405338287531387790158613543822, 10.47489470649990264121942926289, 11.39084427309483692159331220442, 12.10086058890517263052992663818

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