↑

Properties

Label 2-195-65.47-c1-0-1
Degree $2$
Conductor $195$
Sign $-0.309 + 0.950i$
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  + 2.48i·2-s + (−0.707 + 0.707i)3-s − 4.18·4-s + (−2.00 − 0.999i)5-s + (−1.75 − 1.75i)6-s − 0.242·7-s − 5.43i·8-s − 1.00i·9-s + (2.48 − 4.97i)10-s + (−4.24 + 4.24i)11-s + (2.95 − 2.95i)12-s + (2.81 − 2.25i)13-s − 0.602i·14-s + (2.12 − 0.707i)15-s + 5.13·16-s + (−1.37 + 1.37i)17-s + ⋯
L(s)  = 1  + 1.75i·2-s + (−0.408 + 0.408i)3-s − 2.09·4-s + (−0.894 − 0.446i)5-s + (−0.717 − 0.717i)6-s − 0.0916·7-s − 1.92i·8-s − 0.333i·9-s + (0.785 − 1.57i)10-s + (−1.27 + 1.27i)11-s + (0.854 − 0.854i)12-s + (0.781 − 0.624i)13-s − 0.161i·14-s + (0.547 − 0.182i)15-s + 1.28·16-s + (−0.332 + 0.332i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.309 + 0.950i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.309 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.239979 - 0.330592i\)
\(L(\frac12)\) \(\approx\) \(0.239979 - 0.330592i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (2.00 + 0.999i)T \)
13 \( 1 + (-2.81 + 2.25i)T \)
good2 \( 1 - 2.48iT - 2T^{2} \)
7 \( 1 + 0.242T + 7T^{2} \)
11 \( 1 + (4.24 - 4.24i)T - 11iT^{2} \)
17 \( 1 + (1.37 - 1.37i)T - 17iT^{2} \)
19 \( 1 + (3.91 - 3.91i)T - 19iT^{2} \)
23 \( 1 + (-1.90 - 1.90i)T + 23iT^{2} \)
29 \( 1 - 5.76iT - 29T^{2} \)
31 \( 1 + (5.69 + 5.69i)T + 31iT^{2} \)
37 \( 1 - 3.31T + 37T^{2} \)
41 \( 1 + (1.51 + 1.51i)T + 41iT^{2} \)
43 \( 1 + (-3.88 - 3.88i)T + 43iT^{2} \)
47 \( 1 + 1.68T + 47T^{2} \)
53 \( 1 + (2.22 - 2.22i)T - 53iT^{2} \)
59 \( 1 + (-5.27 - 5.27i)T + 59iT^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 - 15.3iT - 67T^{2} \)
71 \( 1 + (0.0780 + 0.0780i)T + 71iT^{2} \)
73 \( 1 - 1.45iT - 73T^{2} \)
79 \( 1 + 7.60iT - 79T^{2} \)
83 \( 1 - 2.71T + 83T^{2} \)
89 \( 1 + (-0.887 - 0.887i)T + 89iT^{2} \)
97 \( 1 + 4.76iT - 97T^{2} \)
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

−13.00784699569054191555259438240, −12.77725366916193683326392780868, −11.13218968870089652457241007416, −10.02054456493659992855015744756, −8.792007695965053719856700101261, −7.949349469895714363860402951360, −7.16673368960315730066481296595, −5.85618432690710847672025782313, −4.95223987701946626260793089533, −3.97016001481232995397574570989, 0.37382902789496502765458569476, 2.49386638560767244758418997262, 3.63177331576132366446158940391, 4.92249580958428651594851724188, 6.56493400106427223071986538457, 8.101201679564635671304293377383, 8.950167452440598844323990538332, 10.45135022614234513822978356940, 11.15240843467942378312147180031, 11.42435264142577101021195700038

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