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Properties

Label 2-195-65.64-c1-0-15
Degree $2$
Conductor $195$
Sign $-0.986 + 0.165i$
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  + 0.246·2-s − i·3-s − 1.93·4-s + (−2.15 − 0.608i)5-s − 0.246i·6-s − 2.59·7-s − 0.971·8-s − 9-s + (−0.530 − 0.150i)10-s + 3.81i·11-s + 1.93i·12-s + (−1.54 − 3.25i)13-s − 0.639·14-s + (−0.608 + 2.15i)15-s + 3.63·16-s − 4.63i·17-s + ⋯
L(s)  = 1  + 0.174·2-s − 0.577i·3-s − 0.969·4-s + (−0.962 − 0.272i)5-s − 0.100i·6-s − 0.979·7-s − 0.343·8-s − 0.333·9-s + (−0.167 − 0.0474i)10-s + 1.14i·11-s + 0.559i·12-s + (−0.427 − 0.903i)13-s − 0.170·14-s + (−0.157 + 0.555i)15-s + 0.909·16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 + 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.986 + 0.165i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.986 + 0.165i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0239857 - 0.287557i\)
\(L(\frac12)\) \(\approx\) \(0.0239857 - 0.287557i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (2.15 + 0.608i)T \)
13 \( 1 + (1.54 + 3.25i)T \)
good2 \( 1 - 0.246T + 2T^{2} \)
7 \( 1 + 2.59T + 7T^{2} \)
11 \( 1 - 3.81iT - 11T^{2} \)
17 \( 1 + 4.63iT - 17T^{2} \)
19 \( 1 + 5.02iT - 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 7.87T + 29T^{2} \)
31 \( 1 + 10.2iT - 31T^{2} \)
37 \( 1 + 4.53T + 37T^{2} \)
41 \( 1 - 6.40iT - 41T^{2} \)
43 \( 1 - 0.639iT - 43T^{2} \)
47 \( 1 - 3.81T + 47T^{2} \)
53 \( 1 - 6.51iT - 53T^{2} \)
59 \( 1 - 6.24iT - 59T^{2} \)
61 \( 1 + 3.23T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + 3.08T + 73T^{2} \)
79 \( 1 + 3.36T + 79T^{2} \)
83 \( 1 + 7.23T + 83T^{2} \)
89 \( 1 + 8.83iT - 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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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

−12.34846895996563841482115339282, −11.34354369587199063864792090265, −9.753174043955126183800867909320, −9.181438762184720477156466457899, −7.79378105350331955494126037255, −7.13470457099462182746199653959, −5.50557026114814568919629301587, −4.39105335781081027316860599224, −3.05020600305848268408836472585, −0.24066739436366115743729805218, 3.44615993875452634495623950968, 3.98601210631549390736197767244, 5.45946799763908468772331191280, 6.68551376558610439354003748091, 8.247015366183810385574709466810, 8.917307991142103877336346166864, 10.04482077547915074937783251073, 10.90523884591912823167373701676, 12.16283146001152019237518409519, 12.83582592859793970127481294333

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