Properties

Label 2-195-3.2-c2-0-18
Degree $2$
Conductor $195$
Sign $0.181 + 0.983i$
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  − 0.405i·2-s + (−2.95 + 0.545i)3-s + 3.83·4-s − 2.23i·5-s + (0.221 + 1.19i)6-s − 4.55·7-s − 3.18i·8-s + (8.40 − 3.21i)9-s − 0.907·10-s − 6.04i·11-s + (−11.3 + 2.09i)12-s + 3.60·13-s + 1.84i·14-s + (1.21 + 6.59i)15-s + 14.0·16-s − 30.9i·17-s + ⋯
L(s)  = 1  − 0.202i·2-s + (−0.983 + 0.181i)3-s + 0.958·4-s − 0.447i·5-s + (0.0368 + 0.199i)6-s − 0.651·7-s − 0.397i·8-s + (0.933 − 0.357i)9-s − 0.0907·10-s − 0.549i·11-s + (−0.942 + 0.174i)12-s + 0.277·13-s + 0.132i·14-s + (0.0812 + 0.439i)15-s + 0.878·16-s − 1.81i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.181 + 0.983i$
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: \((2,\ 195,\ (\ :1),\ 0.181 + 0.983i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.929774 - 0.773662i\)
\(L(\frac12)\) \(\approx\) \(0.929774 - 0.773662i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.95 - 0.545i)T \)
5 \( 1 + 2.23iT \)
13 \( 1 - 3.60T \)
good2 \( 1 + 0.405iT - 4T^{2} \)
7 \( 1 + 4.55T + 49T^{2} \)
11 \( 1 + 6.04iT - 121T^{2} \)
17 \( 1 + 30.9iT - 289T^{2} \)
19 \( 1 - 0.389T + 361T^{2} \)
23 \( 1 + 14.8iT - 529T^{2} \)
29 \( 1 + 49.2iT - 841T^{2} \)
31 \( 1 - 30.8T + 961T^{2} \)
37 \( 1 - 22.9T + 1.36e3T^{2} \)
41 \( 1 - 41.8iT - 1.68e3T^{2} \)
43 \( 1 + 36.5T + 1.84e3T^{2} \)
47 \( 1 - 74.9iT - 2.20e3T^{2} \)
53 \( 1 - 38.0iT - 2.80e3T^{2} \)
59 \( 1 - 103. iT - 3.48e3T^{2} \)
61 \( 1 - 7.55T + 3.72e3T^{2} \)
67 \( 1 - 70.0T + 4.48e3T^{2} \)
71 \( 1 + 75.9iT - 5.04e3T^{2} \)
73 \( 1 + 4.52T + 5.32e3T^{2} \)
79 \( 1 + 59.3T + 6.24e3T^{2} \)
83 \( 1 - 41.2iT - 6.88e3T^{2} \)
89 \( 1 + 85.8iT - 7.92e3T^{2} \)
97 \( 1 - 182.T + 9.40e3T^{2} \)
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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

−11.77978065203740223768957033914, −11.35992701113986692356957096906, −10.19563593916307689753263822263, −9.429101568451484527196936810487, −7.79824770015142847754273400068, −6.61131996950712195125581199021, −5.90231130018192308646916096629, −4.51771240543777074614320557262, −2.88142839278010796188485495177, −0.802191974006879788138725202235, 1.77993670808982082749379518655, 3.60234466384131275664627652957, 5.39219570530274105913660305652, 6.44391739706148192480795314583, 6.95346483955408686345409614962, 8.185197951492547387416768488683, 9.962429007783753106628728627489, 10.61440697101920268253847476024, 11.47608605062154621748015817242, 12.41313294634448386231679829095

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