Properties

Label 2-195-13.3-c1-0-9
Degree $2$
Conductor $195$
Sign $0.0128 - 0.999i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−0.5 − 0.866i)3-s + (−0.999 + 1.73i)4-s − 5-s + (−0.999 + 1.73i)6-s + (−2.5 + 4.33i)7-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)10-s + (−1 − 1.73i)11-s + 1.99·12-s + (−2.5 − 2.59i)13-s + 10·14-s + (0.5 + 0.866i)15-s + (1.99 + 3.46i)16-s + (−1 + 1.73i)17-s + 1.99·18-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.288 − 0.499i)3-s + (−0.499 + 0.866i)4-s − 0.447·5-s + (−0.408 + 0.707i)6-s + (−0.944 + 1.63i)7-s + (−0.166 + 0.288i)9-s + (0.316 + 0.547i)10-s + (−0.301 − 0.522i)11-s + 0.577·12-s + (−0.693 − 0.720i)13-s + 2.67·14-s + (0.129 + 0.223i)15-s + (0.499 + 0.866i)16-s + (−0.242 + 0.420i)17-s + 0.471·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + T \)
13 \( 1 + (2.5 + 2.59i)T \)
good2 \( 1 + (1 + 1.73i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (2.5 - 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 + 11.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6 - 10.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 15T + 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.5 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.83324871917768837763056319596, −10.86433363904438056000190268925, −9.917225851315903577146784462859, −8.874883812046108081362344593792, −8.161078769068842431841901569056, −6.48950908792604487179936868155, −5.44591239539140253733956030522, −3.25549513136558277618457528319, −2.28099371756223378377514074296, 0, 3.60602569555253128313913909954, 4.85195401693137763199433100987, 6.40863876029563659747050316072, 7.17698900774578246291183003390, 7.88490645501255769625221699026, 9.487848919369454543324719087155, 9.820408264263129384923454345892, 11.04285154762065123826130800326, 12.22903265456050207221478858935

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