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Properties

Label 2-195-13.4-c1-0-1
Degree $2$
Conductor $195$
Sign $-0.00197 - 0.999i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.78 + 1.02i)2-s + (−0.5 + 0.866i)3-s + (1.11 + 1.93i)4-s + i·5-s + (−1.78 + 1.02i)6-s + (−2.01 + 1.16i)7-s + 0.492i·8-s + (−0.499 − 0.866i)9-s + (−1.02 + 1.78i)10-s + (4.30 + 2.48i)11-s − 2.23·12-s + (3.14 − 1.76i)13-s − 4.79·14-s + (−0.866 − 0.5i)15-s + (1.73 − 3.00i)16-s + (−3.84 − 6.65i)17-s + ⋯
L(s)  = 1  + (1.26 + 0.727i)2-s + (−0.288 + 0.499i)3-s + (0.559 + 0.969i)4-s + 0.447i·5-s + (−0.727 + 0.420i)6-s + (−0.761 + 0.439i)7-s + 0.174i·8-s + (−0.166 − 0.288i)9-s + (−0.325 + 0.563i)10-s + (1.29 + 0.749i)11-s − 0.646·12-s + (0.871 − 0.490i)13-s − 1.28·14-s + (−0.223 − 0.129i)15-s + (0.433 − 0.750i)16-s + (−0.931 − 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00197 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00197 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.00197 - 0.999i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.00197 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37565 + 1.37836i\)
\(L(\frac12)\) \(\approx\) \(1.37565 + 1.37836i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
13 \( 1 + (-3.14 + 1.76i)T \)
good2 \( 1 + (-1.78 - 1.02i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (2.01 - 1.16i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.30 - 2.48i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.84 + 6.65i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.50 - 1.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.81 + 4.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.48 - 4.30i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.61iT - 31T^{2} \)
37 \( 1 + (-0.566 - 0.326i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.16 + 2.40i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.995 - 1.72i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.5iT - 47T^{2} \)
53 \( 1 + 9.94T + 53T^{2} \)
59 \( 1 + (3.25 - 1.88i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.52 - 2.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.04 - 4.64i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.71 - 0.990i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.601iT - 73T^{2} \)
79 \( 1 - 1.62T + 79T^{2} \)
83 \( 1 - 11.9iT - 83T^{2} \)
89 \( 1 + (5.23 + 3.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.5 - 8.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.82791175506885442870942641636, −12.08881802169427137140404306706, −11.00664049424795641100412530958, −9.770590475985436263398010425830, −8.826023827410780869915092082564, −6.87257817534558157326919330663, −6.56164451438105462648833690953, −5.30683147826190400768159076856, −4.20783287384807015759157919705, −3.10207509527095614377946243348, 1.63743867874231517256293967472, 3.54935340176711841996376416632, 4.32026560349811705761911067599, 6.00178374470368637135694413822, 6.47034361346541486664410870489, 8.281478821419639835487793313709, 9.366655629699051300260324241620, 10.98906036060474999449177322608, 11.35730617995637992488493029329, 12.48091677608735053303811903461

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