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Properties

Label 2-195-39.5-c1-0-11
Degree $2$
Conductor $195$
Sign $0.407 + 0.913i$
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  + (−0.260 − 0.260i)2-s + (1.18 − 1.26i)3-s − 1.86i·4-s + (0.707 + 0.707i)5-s + (−0.637 + 0.0199i)6-s + (2.54 + 2.54i)7-s + (−1.00 + 1.00i)8-s + (−0.187 − 2.99i)9-s − 0.368i·10-s + (−0.348 + 0.348i)11-s + (−2.35 − 2.21i)12-s + (−0.339 − 3.58i)13-s − 1.32i·14-s + (1.73 − 0.0541i)15-s − 3.20·16-s − 5.28·17-s + ⋯
L(s)  = 1  + (−0.184 − 0.184i)2-s + (0.684 − 0.728i)3-s − 0.932i·4-s + (0.316 + 0.316i)5-s + (−0.260 + 0.00814i)6-s + (0.961 + 0.961i)7-s + (−0.355 + 0.355i)8-s + (−0.0625 − 0.998i)9-s − 0.116i·10-s + (−0.105 + 0.105i)11-s + (−0.679 − 0.638i)12-s + (−0.0942 − 0.995i)13-s − 0.354i·14-s + (0.446 − 0.0139i)15-s − 0.801·16-s − 1.28·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.407 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.407 + 0.913i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.407 + 0.913i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18708 - 0.770213i\)
\(L(\frac12)\) \(\approx\) \(1.18708 - 0.770213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.18 + 1.26i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (0.339 + 3.58i)T \)
good2 \( 1 + (0.260 + 0.260i)T + 2iT^{2} \)
7 \( 1 + (-2.54 - 2.54i)T + 7iT^{2} \)
11 \( 1 + (0.348 - 0.348i)T - 11iT^{2} \)
17 \( 1 + 5.28T + 17T^{2} \)
19 \( 1 + (3.44 - 3.44i)T - 19iT^{2} \)
23 \( 1 - 9.38T + 23T^{2} \)
29 \( 1 - 6.80iT - 29T^{2} \)
31 \( 1 + (2.96 - 2.96i)T - 31iT^{2} \)
37 \( 1 + (-1.76 - 1.76i)T + 37iT^{2} \)
41 \( 1 + (-2.21 - 2.21i)T + 41iT^{2} \)
43 \( 1 + 5.41iT - 43T^{2} \)
47 \( 1 + (-2.46 + 2.46i)T - 47iT^{2} \)
53 \( 1 - 6.94iT - 53T^{2} \)
59 \( 1 + (-0.248 + 0.248i)T - 59iT^{2} \)
61 \( 1 + 3.60T + 61T^{2} \)
67 \( 1 + (1.34 - 1.34i)T - 67iT^{2} \)
71 \( 1 + (11.6 + 11.6i)T + 71iT^{2} \)
73 \( 1 + (-6.48 - 6.48i)T + 73iT^{2} \)
79 \( 1 - 2.66T + 79T^{2} \)
83 \( 1 + (7.35 + 7.35i)T + 83iT^{2} \)
89 \( 1 + (-2.54 + 2.54i)T - 89iT^{2} \)
97 \( 1 + (-10.4 + 10.4i)T - 97iT^{2} \)
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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

−12.42891614522693425489623266767, −11.21091396543292436988888244747, −10.47560735539726981016521232153, −9.049257236494285819560524394767, −8.613170807874800633061355248584, −7.21052508270978886150611833669, −6.06310260449751041722061015947, −5.00595975340591869973887717780, −2.75678161822269981923338138152, −1.65024306273990765320776425125, 2.40532920127720945895066169821, 4.12530720317598861504824902769, 4.70345180033736610912440510967, 6.82207841948882328646823254427, 7.78520277778224233576398649811, 8.786084637668079179383462902410, 9.350446577441500237477032599004, 10.87577166230039027190439101503, 11.38155239391739082996179613291, 13.09184427334290255740308150539

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