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Properties

Label 2-195-65.49-c1-0-13
Degree $2$
Conductor $195$
Sign $0.0906 + 0.995i$
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  + (−0.296 − 0.512i)2-s + (0.866 − 0.5i)3-s + (0.824 − 1.42i)4-s + (0.903 − 2.04i)5-s + (−0.512 − 0.296i)6-s + (−0.828 + 1.43i)7-s − 2.16·8-s + (0.499 − 0.866i)9-s + (−1.31 + 0.142i)10-s + (−1.22 + 0.706i)11-s − 1.64i·12-s + (−0.0145 + 3.60i)13-s + 0.981·14-s + (−0.240 − 2.22i)15-s + (−1.00 − 1.74i)16-s + (−2.83 − 1.63i)17-s + ⋯
L(s)  = 1  + (−0.209 − 0.362i)2-s + (0.499 − 0.288i)3-s + (0.412 − 0.714i)4-s + (0.404 − 0.914i)5-s + (−0.209 − 0.120i)6-s + (−0.313 + 0.542i)7-s − 0.763·8-s + (0.166 − 0.288i)9-s + (−0.416 + 0.0449i)10-s + (−0.368 + 0.212i)11-s − 0.476i·12-s + (−0.00404 + 0.999i)13-s + 0.262·14-s + (−0.0620 − 0.574i)15-s + (−0.252 − 0.437i)16-s + (−0.688 − 0.397i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0906 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0906 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.0906 + 0.995i$
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: \((2,\ 195,\ (\ :1/2),\ 0.0906 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.992524 - 0.906261i\)
\(L(\frac12)\) \(\approx\) \(0.992524 - 0.906261i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.903 + 2.04i)T \)
13 \( 1 + (0.0145 - 3.60i)T \)
good2 \( 1 + (0.296 + 0.512i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.828 - 1.43i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.22 - 0.706i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.83 + 1.63i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.29 - 4.21i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.70 + 3.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.252 - 0.438i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.791iT - 31T^{2} \)
37 \( 1 + (-2.37 - 4.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.67 - 4.42i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.93 - 1.11i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 3.43T + 47T^{2} \)
53 \( 1 + 0.422iT - 53T^{2} \)
59 \( 1 + (-11.3 - 6.54i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.463 - 0.803i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.21 + 10.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.947 - 0.547i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.2T + 73T^{2} \)
79 \( 1 + 9.25T + 79T^{2} \)
83 \( 1 + 4.02T + 83T^{2} \)
89 \( 1 + (0.517 - 0.299i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.57 + 2.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.17942284896475324307702676665, −11.47736530305452632732046302976, −10.02122139575806435715090653705, −9.376826138804157389510456232908, −8.590058027098881498512582872919, −7.09306153154870994544744245103, −5.95644188157715376721195910935, −4.82772929778326199332664872094, −2.78919399354917932858460084094, −1.46411652457451733404275600634, 2.74990210098022556449518027939, 3.52442262825384246060585845195, 5.48768109609770698919895088379, 6.93835771909497075079237280382, 7.45734101804623118865185287483, 8.665775515817689954441270464473, 9.747979858231678543776270392529, 10.73689511981265139291477961500, 11.55389484068843406952162974415, 13.09507547906979370926616800133

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