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Properties

Label 2-195-65.49-c1-0-0
Degree $2$
Conductor $195$
Sign $0.790 - 0.612i$
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  + (−1.36 − 2.36i)2-s + (0.866 − 0.5i)3-s + (−2.72 + 4.71i)4-s + (−1.91 + 1.16i)5-s + (−2.36 − 1.36i)6-s + (−1.86 + 3.23i)7-s + 9.39·8-s + (0.499 − 0.866i)9-s + (5.35 + 2.92i)10-s + (−2.02 + 1.17i)11-s + 5.44i·12-s + (−1.39 + 3.32i)13-s + 10.1·14-s + (−1.07 + 1.96i)15-s + (−7.36 − 12.7i)16-s + (−2.29 − 1.32i)17-s + ⋯
L(s)  = 1  + (−0.964 − 1.67i)2-s + (0.499 − 0.288i)3-s + (−1.36 + 2.35i)4-s + (−0.854 + 0.519i)5-s + (−0.964 − 0.556i)6-s + (−0.705 + 1.22i)7-s + 3.32·8-s + (0.166 − 0.288i)9-s + (1.69 + 0.926i)10-s + (−0.611 + 0.352i)11-s + 1.57i·12-s + (−0.386 + 0.922i)13-s + 2.72·14-s + (−0.277 + 0.506i)15-s + (−1.84 − 3.19i)16-s + (−0.557 − 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.790 - 0.612i$
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: \((2,\ 195,\ (\ :1/2),\ 0.790 - 0.612i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.325208 + 0.111331i\)
\(L(\frac12)\) \(\approx\) \(0.325208 + 0.111331i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.91 - 1.16i)T \)
13 \( 1 + (1.39 - 3.32i)T \)
good2 \( 1 + (1.36 + 2.36i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1.86 - 3.23i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.02 - 1.17i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.29 + 1.32i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.93 + 1.11i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.387 - 0.223i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.774 - 1.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.17iT - 31T^{2} \)
37 \( 1 + (0.797 + 1.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.866 - 0.500i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.1 - 5.83i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.06T + 47T^{2} \)
53 \( 1 - 8.33iT - 53T^{2} \)
59 \( 1 + (4.61 + 2.66i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.317 + 0.550i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.06 - 8.76i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.28 - 1.89i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.0493T + 73T^{2} \)
79 \( 1 - 1.93T + 79T^{2} \)
83 \( 1 - 7.63T + 83T^{2} \)
89 \( 1 + (4.56 - 2.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.24 + 10.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.36893617234613478057563699769, −11.62635711660281566394044189492, −10.72691869609103212600721134598, −9.578889294629104391505611711035, −8.929342117299626281652239913859, −8.002228656280652045670796339276, −6.90044076684386040525216645027, −4.41943092147996038069990494050, −3.04919533655644034722439817071, −2.28401247754001884695302363036, 0.38076046953745897754166192158, 3.94025202939109163413401020038, 5.11606926990621635402554399079, 6.51982872583992419699170563758, 7.62710361099660349935430785247, 8.080386999625088964865026234790, 9.110722896319194274724904756302, 10.13569102395315530309669746143, 10.78803088895285907587684946036, 12.90276086671931134844298894404

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