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Properties

Label 2-195-3.2-c2-0-25
Degree $2$
Conductor $195$
Sign $0.206 + 0.978i$
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  − 1.11i·2-s + (2.93 − 0.620i)3-s + 2.74·4-s − 2.23i·5-s + (−0.694 − 3.28i)6-s − 5.59·7-s − 7.55i·8-s + (8.23 − 3.64i)9-s − 2.50·10-s − 6.16i·11-s + (8.06 − 1.70i)12-s − 3.60·13-s + 6.26i·14-s + (−1.38 − 6.56i)15-s + 2.53·16-s + 15.1i·17-s + ⋯
L(s)  = 1  − 0.559i·2-s + (0.978 − 0.206i)3-s + 0.686·4-s − 0.447i·5-s + (−0.115 − 0.547i)6-s − 0.799·7-s − 0.944i·8-s + (0.914 − 0.404i)9-s − 0.250·10-s − 0.560i·11-s + (0.671 − 0.142i)12-s − 0.277·13-s + 0.447i·14-s + (−0.0924 − 0.437i)15-s + 0.158·16-s + 0.889i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.206 + 0.978i$
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: \((2,\ 195,\ (\ :1),\ 0.206 + 0.978i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.78902 - 1.45041i\)
\(L(\frac12)\) \(\approx\) \(1.78902 - 1.45041i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.93 + 0.620i)T \)
5 \( 1 + 2.23iT \)
13 \( 1 + 3.60T \)
good2 \( 1 + 1.11iT - 4T^{2} \)
7 \( 1 + 5.59T + 49T^{2} \)
11 \( 1 + 6.16iT - 121T^{2} \)
17 \( 1 - 15.1iT - 289T^{2} \)
19 \( 1 - 20.3T + 361T^{2} \)
23 \( 1 - 33.6iT - 529T^{2} \)
29 \( 1 + 7.19iT - 841T^{2} \)
31 \( 1 + 24.0T + 961T^{2} \)
37 \( 1 - 22.3T + 1.36e3T^{2} \)
41 \( 1 + 21.1iT - 1.68e3T^{2} \)
43 \( 1 + 43.3T + 1.84e3T^{2} \)
47 \( 1 - 76.6iT - 2.20e3T^{2} \)
53 \( 1 - 85.5iT - 2.80e3T^{2} \)
59 \( 1 + 16.4iT - 3.48e3T^{2} \)
61 \( 1 - 65.6T + 3.72e3T^{2} \)
67 \( 1 - 46.7T + 4.48e3T^{2} \)
71 \( 1 - 91.1iT - 5.04e3T^{2} \)
73 \( 1 + 82.8T + 5.32e3T^{2} \)
79 \( 1 - 60.9T + 6.24e3T^{2} \)
83 \( 1 + 106. iT - 6.88e3T^{2} \)
89 \( 1 - 89.3iT - 7.92e3T^{2} \)
97 \( 1 + 90.4T + 9.40e3T^{2} \)
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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

−12.21279463413945707794287122463, −11.14977006874387216389596310774, −9.905159941464048339960724778242, −9.321082301825348558994631809568, −8.002470341190423191700170578592, −7.07310147835276460916892098330, −5.84512889723837329082574467647, −3.82140188010818864902158193770, −2.92638020182767399288906504102, −1.41286844786832100477217044361, 2.31605783868691322310324833460, 3.34693528117161621876107472378, 5.06804468986084546156327193881, 6.66789465067635840969925208741, 7.23255475797952360010828157749, 8.297959306049896450514776542391, 9.549096302318881803795013162928, 10.26787463066673924475330375083, 11.50864934761133820768724264638, 12.60571047132436617075928217759

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