↑

Properties

Label 2-195-195.59-c1-0-11
Degree $2$
Conductor $195$
Sign $0.772 - 0.634i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.25 − 0.335i)2-s + (0.377 + 1.69i)3-s + (−0.275 + 0.159i)4-s + (2.14 + 0.638i)5-s + (1.04 + 1.99i)6-s + (0.589 − 2.20i)7-s + (−2.12 + 2.12i)8-s + (−2.71 + 1.27i)9-s + (2.89 + 0.0806i)10-s + (−2.74 + 0.735i)11-s + (−0.373 − 0.406i)12-s + (2.68 − 2.40i)13-s − 2.95i·14-s + (−0.270 + 3.86i)15-s + (−1.63 + 2.82i)16-s + (3.93 − 2.27i)17-s + ⋯
L(s)  = 1  + (0.885 − 0.237i)2-s + (0.218 + 0.975i)3-s + (−0.137 + 0.0796i)4-s + (0.958 + 0.285i)5-s + (0.424 + 0.812i)6-s + (0.222 − 0.832i)7-s + (−0.751 + 0.751i)8-s + (−0.904 + 0.425i)9-s + (0.916 + 0.0254i)10-s + (−0.827 + 0.221i)11-s + (−0.107 − 0.117i)12-s + (0.744 − 0.667i)13-s − 0.789i·14-s + (−0.0697 + 0.997i)15-s + (−0.407 + 0.706i)16-s + (0.953 − 0.550i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.772 - 0.634i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.772 - 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78171 + 0.637698i\)
\(L(\frac12)\) \(\approx\) \(1.78171 + 0.637698i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.377 - 1.69i)T \)
5 \( 1 + (-2.14 - 0.638i)T \)
13 \( 1 + (-2.68 + 2.40i)T \)
good2 \( 1 + (-1.25 + 0.335i)T + (1.73 - i)T^{2} \)
7 \( 1 + (-0.589 + 2.20i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.74 - 0.735i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-3.93 + 2.27i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.775 + 2.89i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.387 - 0.223i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.98 + 2.87i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.62 + 4.62i)T + 31iT^{2} \)
37 \( 1 + (10.2 - 2.75i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.217 - 0.811i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.38 - 4.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.06 - 3.06i)T - 47iT^{2} \)
53 \( 1 - 8.28T + 53T^{2} \)
59 \( 1 + (-1.67 + 6.25i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.66 - 9.80i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.40 - 8.96i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (12.8 + 3.43i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.56 - 1.56i)T + 73iT^{2} \)
79 \( 1 + 9.74T + 79T^{2} \)
83 \( 1 + (-0.856 - 0.856i)T + 83iT^{2} \)
89 \( 1 + (-1.64 + 0.440i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.80 - 2.62i)T + (84.0 + 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.07942262977707583713949417696, −11.53444635816652118096530559189, −10.64187694212427557797125724953, −9.861446220261471190199674529871, −8.810570983491601653429436353950, −7.57044549628788159810865815004, −5.73421492081057515559143822085, −5.08854956479440973791368032756, −3.80065440746131196544262052694, −2.74234985937629016336885823425, 1.79082672062923994213284762575, 3.42897022246690885643645074497, 5.47641225334106078507603662245, 5.68383680092217996469842173130, 6.95604244091053927338082531021, 8.488967178038868618339876945842, 9.131542466070816198017609683662, 10.47216564033505292816411510441, 11.98663737175746933701222791987, 12.66633406935809435715582002850

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