↑

Properties

Label 2-195-13.3-c1-0-2
Degree $2$
Conductor $195$
Sign $-0.320 - 0.947i$
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  + (0.837 + 1.45i)2-s + (0.5 + 0.866i)3-s + (−0.403 + 0.698i)4-s − 5-s + (−0.837 + 1.45i)6-s + (−1.81 + 3.15i)7-s + 2·8-s + (−0.499 + 0.866i)9-s + (−0.837 − 1.45i)10-s + (−0.806 − 1.39i)11-s − 0.806·12-s + (3.22 + 1.61i)13-s − 6.09·14-s + (−0.5 − 0.866i)15-s + (2.48 + 4.29i)16-s + (1.64 − 2.84i)17-s + ⋯
L(s)  = 1  + (0.592 + 1.02i)2-s + (0.288 + 0.499i)3-s + (−0.201 + 0.349i)4-s − 0.447·5-s + (−0.341 + 0.592i)6-s + (−0.687 + 1.19i)7-s + 0.707·8-s + (−0.166 + 0.288i)9-s + (−0.264 − 0.458i)10-s + (−0.243 − 0.420i)11-s − 0.232·12-s + (0.893 + 0.448i)13-s − 1.62·14-s + (−0.129 − 0.223i)15-s + (0.620 + 1.07i)16-s + (0.398 − 0.690i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.320 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.320 - 0.947i$
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: \((2,\ 195,\ (\ :1/2),\ -0.320 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.952411 + 1.32766i\)
\(L(\frac12)\) \(\approx\) \(0.952411 + 1.32766i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.22 - 1.61i)T \)
good2 \( 1 + (-0.837 - 1.45i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (1.81 - 3.15i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.806 + 1.39i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.64 + 2.84i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.55 + 6.16i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.86 + 4.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.72 + 6.44i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.76T + 31T^{2} \)
37 \( 1 + (-5.96 - 10.3i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.56 - 4.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.54 + 6.14i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.67T + 47T^{2} \)
53 \( 1 + 1.42T + 53T^{2} \)
59 \( 1 + (5.78 - 10.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 - 5.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.70 + 4.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.91 + 3.31i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.67T + 73T^{2} \)
79 \( 1 - 2.45T + 79T^{2} \)
83 \( 1 + 2.96T + 83T^{2} \)
89 \( 1 + (0.590 + 1.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.50 - 7.80i)T + (-48.5 - 84.0i)T^{2} \)
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

−13.18736997780637652616935965071, −11.87353160823432158307736875586, −10.95725144459026099226699917122, −9.584416400973523905079990211227, −8.706530046802513015888523725856, −7.61267071116931005603988460364, −6.37131386482780679721109248019, −5.53245073645329760536780092466, −4.34914335822642715026035122329, −2.89084079680836657214642924932, 1.49398246950874448614475369033, 3.44719439313873068612829143730, 3.86420377216612124392020250887, 5.73757774514313004784633224126, 7.36566725036637731186176188693, 7.83063776018202863584523239713, 9.569027652745587510271919660806, 10.56303015582400129925655811122, 11.26451912680982002810510275105, 12.64088513690278590287058860132

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