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Properties

Label 2-195-3.2-c2-0-7
Degree $2$
Conductor $195$
Sign $-0.972 + 0.233i$
Analytic cond. $5.31336$
Root an. cond. $2.30507$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.37i·2-s + (0.699 + 2.91i)3-s − 1.65·4-s + 2.23i·5-s + (−6.93 + 1.66i)6-s − 2.88·7-s + 5.58i·8-s + (−8.02 + 4.08i)9-s − 5.31·10-s − 11.6i·11-s + (−1.15 − 4.81i)12-s + 3.60·13-s − 6.85i·14-s + (−6.52 + 1.56i)15-s − 19.8·16-s + 33.5i·17-s + ⋯
L(s)  = 1  + 1.18i·2-s + (0.233 + 0.972i)3-s − 0.412·4-s + 0.447i·5-s + (−1.15 + 0.277i)6-s − 0.412·7-s + 0.697i·8-s + (−0.891 + 0.453i)9-s − 0.531·10-s − 1.06i·11-s + (−0.0962 − 0.401i)12-s + 0.277·13-s − 0.489i·14-s + (−0.434 + 0.104i)15-s − 1.24·16-s + 1.97i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.972 + 0.233i$
Analytic conductor: \(5.31336\)
Root analytic conductor: \(2.30507\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1),\ -0.972 + 0.233i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.175607 - 1.48502i\)
\(L(\frac12)\) \(\approx\) \(0.175607 - 1.48502i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.699 - 2.91i)T \)
5 \( 1 - 2.23iT \)
13 \( 1 - 3.60T \)
good2 \( 1 - 2.37iT - 4T^{2} \)
7 \( 1 + 2.88T + 49T^{2} \)
11 \( 1 + 11.6iT - 121T^{2} \)
17 \( 1 - 33.5iT - 289T^{2} \)
19 \( 1 - 17.9T + 361T^{2} \)
23 \( 1 + 15.3iT - 529T^{2} \)
29 \( 1 + 36.8iT - 841T^{2} \)
31 \( 1 - 20.8T + 961T^{2} \)
37 \( 1 - 12.9T + 1.36e3T^{2} \)
41 \( 1 - 68.5iT - 1.68e3T^{2} \)
43 \( 1 - 61.7T + 1.84e3T^{2} \)
47 \( 1 - 68.7iT - 2.20e3T^{2} \)
53 \( 1 + 47.7iT - 2.80e3T^{2} \)
59 \( 1 - 84.7iT - 3.48e3T^{2} \)
61 \( 1 - 22.8T + 3.72e3T^{2} \)
67 \( 1 - 29.1T + 4.48e3T^{2} \)
71 \( 1 - 99.3iT - 5.04e3T^{2} \)
73 \( 1 - 18.1T + 5.32e3T^{2} \)
79 \( 1 - 63.3T + 6.24e3T^{2} \)
83 \( 1 + 57.8iT - 6.88e3T^{2} \)
89 \( 1 + 143. iT - 7.92e3T^{2} \)
97 \( 1 - 17.3T + 9.40e3T^{2} \)
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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

−13.18085905048780992810335703714, −11.51992482734201867223257274660, −10.77986334408078355773802570185, −9.726594531408676295049191038018, −8.511240562102314777375848017331, −7.905657523997336808090103529189, −6.29078789650251245807546908150, −5.81858548388043774385384654790, −4.25061180556866435486427507108, −2.90787000814013056446500129576, 0.860470702552565229226813549884, 2.26078494182832516641776875098, 3.43217973846744243819789346832, 5.14406044471798320775489395460, 6.80338504080230596124253092803, 7.52120784724544923985776191213, 9.141207527237414770356263451015, 9.678357528606339460576932139234, 11.05881344055741241629461003816, 12.03170373913822205458311563097

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