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Properties

Label 2-195-13.3-c1-0-4
Degree $2$
Conductor $195$
Sign $0.974 - 0.224i$
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  + (0.169 + 0.294i)2-s + (0.5 + 0.866i)3-s + (0.942 − 1.63i)4-s + 5-s + (−0.169 + 0.294i)6-s + (−0.330 + 0.571i)7-s + 1.32·8-s + (−0.499 + 0.866i)9-s + (0.169 + 0.294i)10-s + (−0.339 − 0.588i)11-s + 1.88·12-s + (1.93 − 3.04i)13-s − 0.224·14-s + (0.5 + 0.866i)15-s + (−1.66 − 2.87i)16-s + (−3.71 + 6.43i)17-s + ⋯
L(s)  = 1  + (0.120 + 0.208i)2-s + (0.288 + 0.499i)3-s + (0.471 − 0.816i)4-s + 0.447·5-s + (−0.0693 + 0.120i)6-s + (−0.124 + 0.216i)7-s + 0.466·8-s + (−0.166 + 0.288i)9-s + (0.0537 + 0.0930i)10-s + (−0.102 − 0.177i)11-s + 0.544·12-s + (0.535 − 0.844i)13-s − 0.0599·14-s + (0.129 + 0.223i)15-s + (−0.415 − 0.718i)16-s + (−0.900 + 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 - 0.224i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.974 - 0.224i$
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: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.974 - 0.224i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54910 + 0.175928i\)
\(L(\frac12)\) \(\approx\) \(1.54910 + 0.175928i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 + 3.04i)T \)
good2 \( 1 + (-0.169 - 0.294i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (0.330 - 0.571i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.339 + 0.588i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.71 - 6.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0577 - 0.100i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.88 - 6.72i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.77 + 4.80i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.97T + 31T^{2} \)
37 \( 1 + (4.88 + 8.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.11 + 3.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.272 + 0.471i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.01T + 47T^{2} \)
53 \( 1 - 0.679T + 53T^{2} \)
59 \( 1 + (1.11 - 1.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.10 + 3.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.81 - 6.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.65 - 6.33i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.01T + 73T^{2} \)
79 \( 1 - 9.97T + 79T^{2} \)
83 \( 1 - 1.76T + 83T^{2} \)
89 \( 1 + (6.77 + 11.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.95 - 8.57i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.79947765340061565335161897300, −11.13000018025538356019998419633, −10.69173995609696905041516149530, −9.626959948319683461082888104850, −8.703585359359659912085758504516, −7.35129735677891017706207058365, −6.00063344088715188927538875985, −5.36961521232894928434742895297, −3.69101048014574956963407307250, −1.96018599622287071952816927742, 2.02846324427282973056080871056, 3.29316567301027369179350186713, 4.80640621050534447526874289051, 6.67094993085870160368038606220, 7.10813651664078485401211453362, 8.499557256236971950349337098081, 9.305537826701512595998351794482, 10.79558917303993318283230998717, 11.55137654159798210265768017459, 12.59173265670982867393079781954

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