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Properties

Label 2-195-195.194-c2-0-7
Degree $2$
Conductor $195$
Sign $-0.555 - 0.831i$
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  + 0.788i·2-s + (−1.53 − 2.57i)3-s + 3.37·4-s + (−1.75 + 4.68i)5-s + (2.03 − 1.21i)6-s − 10.4·7-s + 5.81i·8-s + (−4.28 + 7.91i)9-s + (−3.69 − 1.38i)10-s − 3.29·11-s + (−5.18 − 8.70i)12-s + (5.18 + 11.9i)13-s − 8.27i·14-s + (14.7 − 2.66i)15-s + 8.92·16-s + 5.61·17-s + ⋯
L(s)  = 1  + 0.394i·2-s + (−0.511 − 0.859i)3-s + 0.844·4-s + (−0.350 + 0.936i)5-s + (0.338 − 0.201i)6-s − 1.49·7-s + 0.727i·8-s + (−0.476 + 0.879i)9-s + (−0.369 − 0.138i)10-s − 0.299·11-s + (−0.432 − 0.725i)12-s + (0.398 + 0.917i)13-s − 0.590i·14-s + (0.984 − 0.177i)15-s + 0.558·16-s + 0.330·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(5.31336\)
Root analytic conductor: \(2.30507\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1),\ -0.555 - 0.831i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.379075 + 0.708773i\)
\(L(\frac12)\) \(\approx\) \(0.379075 + 0.708773i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.53 + 2.57i)T \)
5 \( 1 + (1.75 - 4.68i)T \)
13 \( 1 + (-5.18 - 11.9i)T \)
good2 \( 1 - 0.788iT - 4T^{2} \)
7 \( 1 + 10.4T + 49T^{2} \)
11 \( 1 + 3.29T + 121T^{2} \)
17 \( 1 - 5.61T + 289T^{2} \)
19 \( 1 - 22.2iT - 361T^{2} \)
23 \( 1 + 42.7T + 529T^{2} \)
29 \( 1 - 29.0iT - 841T^{2} \)
31 \( 1 + 24.8iT - 961T^{2} \)
37 \( 1 - 35.6T + 1.36e3T^{2} \)
41 \( 1 + 18.1T + 1.68e3T^{2} \)
43 \( 1 - 21.7iT - 1.84e3T^{2} \)
47 \( 1 + 57.8iT - 2.20e3T^{2} \)
53 \( 1 - 68.3T + 2.80e3T^{2} \)
59 \( 1 + 96.0T + 3.48e3T^{2} \)
61 \( 1 + 38.2T + 3.72e3T^{2} \)
67 \( 1 - 3.73T + 4.48e3T^{2} \)
71 \( 1 - 42.6T + 5.04e3T^{2} \)
73 \( 1 - 79.7T + 5.32e3T^{2} \)
79 \( 1 - 61.1T + 6.24e3T^{2} \)
83 \( 1 + 34.8iT - 6.88e3T^{2} \)
89 \( 1 - 20.7T + 7.92e3T^{2} \)
97 \( 1 + 93.0T + 9.40e3T^{2} \)
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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

−12.35428294472877736540789951308, −11.80750163422365974923212768962, −10.76471766581238292864919295545, −9.917926769275999791599586397444, −8.108035906787248751163657860612, −7.26318983736380663255699466267, −6.35944109029156555165932868089, −5.95522993802453378804556310259, −3.53588754186674488754529383273, −2.17708252703554911176721138274, 0.45627712670599490423106574826, 2.98352037464154438660343887622, 4.05362125026857783002364641478, 5.61102984014083750606640752550, 6.45228971969721176716922989568, 7.943370939939833107236160751997, 9.340414541744278575829681223946, 10.04046069051065871969272389321, 10.91397311707398167307989720186, 12.01377903105361672324412902894

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