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Properties

Label 2-195-65.29-c1-0-0
Degree $2$
Conductor $195$
Sign $-0.997 + 0.0647i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.521 + 0.301i)2-s + (−0.866 + 0.5i)3-s + (−0.818 + 1.41i)4-s + (0.446 + 2.19i)5-s + (0.301 − 0.521i)6-s + (−3.08 − 1.77i)7-s − 2.18i·8-s + (0.499 − 0.866i)9-s + (−0.892 − 1.00i)10-s + (−1.30 − 2.26i)11-s − 1.63i·12-s + (−2.88 + 2.16i)13-s + 2.14·14-s + (−1.48 − 1.67i)15-s + (−0.978 − 1.69i)16-s + (1.94 + 1.12i)17-s + ⋯
L(s)  = 1  + (−0.368 + 0.212i)2-s + (−0.499 + 0.288i)3-s + (−0.409 + 0.709i)4-s + (0.199 + 0.979i)5-s + (0.122 − 0.212i)6-s + (−1.16 − 0.672i)7-s − 0.774i·8-s + (0.166 − 0.288i)9-s + (−0.282 − 0.318i)10-s + (−0.394 − 0.682i)11-s − 0.472i·12-s + (−0.800 + 0.599i)13-s + 0.572·14-s + (−0.382 − 0.432i)15-s + (−0.244 − 0.423i)16-s + (0.471 + 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.997 + 0.0647i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.997 + 0.0647i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0108233 - 0.333718i\)
\(L(\frac12)\) \(\approx\) \(0.0108233 - 0.333718i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.446 - 2.19i)T \)
13 \( 1 + (2.88 - 2.16i)T \)
good2 \( 1 + (0.521 - 0.301i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (3.08 + 1.77i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.30 + 2.26i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.94 - 1.12i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.00 - 6.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.43 + 0.826i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.26 - 3.91i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.05T + 31T^{2} \)
37 \( 1 + (-3.64 + 2.10i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.388 - 0.673i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.197 - 0.113i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 3.14iT - 47T^{2} \)
53 \( 1 - 6.42iT - 53T^{2} \)
59 \( 1 + (6.26 - 10.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.46 - 2.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.969 - 0.559i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.66 + 9.81i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.4iT - 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 + (8.60 + 14.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.6 + 6.12i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.89267108028136356282127616528, −12.14558191060167695234807742541, −10.73531134856330569872421048274, −10.11614356537314241858147278104, −9.216930869112527871644387120803, −7.79908470592780523933106350816, −6.88538329244344037066788061340, −5.96734154249175011002197359868, −4.10546336517638567006018251433, −3.13352694602219763492709519297, 0.33705533836138487571464624335, 2.35879034994240260085738238123, 4.76784221095144544975350770783, 5.49794609109358341529124375449, 6.64941103304217496058691379288, 8.161243730477993579804454283075, 9.394532417816689495660484335756, 9.735052884202063253991724787971, 10.92700336881928926427061662862, 12.21092349435531773036346115692

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