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Properties

Label 2-195-39.5-c1-0-12
Degree $2$
Conductor $195$
Sign $0.319 - 0.947i$
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  + (1.57 + 1.57i)2-s + (1.73 + 0.0822i)3-s + 2.98i·4-s + (−0.707 − 0.707i)5-s + (2.60 + 2.86i)6-s + (−2.29 − 2.29i)7-s + (−1.56 + 1.56i)8-s + (2.98 + 0.284i)9-s − 2.23i·10-s + (−3.85 + 3.85i)11-s + (−0.245 + 5.16i)12-s + (−0.766 − 3.52i)13-s − 7.24i·14-s + (−1.16 − 1.28i)15-s + 1.04·16-s − 3.78·17-s + ⋯
L(s)  = 1  + (1.11 + 1.11i)2-s + (0.998 + 0.0474i)3-s + 1.49i·4-s + (−0.316 − 0.316i)5-s + (1.06 + 1.16i)6-s + (−0.866 − 0.866i)7-s + (−0.551 + 0.551i)8-s + (0.995 + 0.0948i)9-s − 0.706i·10-s + (−1.16 + 1.16i)11-s + (−0.0709 + 1.49i)12-s + (−0.212 − 0.977i)13-s − 1.93i·14-s + (−0.300 − 0.330i)15-s + 0.261·16-s − 0.917·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.319 - 0.947i$
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: \((2,\ 195,\ (\ :1/2),\ 0.319 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89577 + 1.36208i\)
\(L(\frac12)\) \(\approx\) \(1.89577 + 1.36208i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 - 0.0822i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (0.766 + 3.52i)T \)
good2 \( 1 + (-1.57 - 1.57i)T + 2iT^{2} \)
7 \( 1 + (2.29 + 2.29i)T + 7iT^{2} \)
11 \( 1 + (3.85 - 3.85i)T - 11iT^{2} \)
17 \( 1 + 3.78T + 17T^{2} \)
19 \( 1 + (1.28 - 1.28i)T - 19iT^{2} \)
23 \( 1 - 5.74T + 23T^{2} \)
29 \( 1 + 5.42iT - 29T^{2} \)
31 \( 1 + (3.95 - 3.95i)T - 31iT^{2} \)
37 \( 1 + (-6.99 - 6.99i)T + 37iT^{2} \)
41 \( 1 + (0.872 + 0.872i)T + 41iT^{2} \)
43 \( 1 - 4.18iT - 43T^{2} \)
47 \( 1 + (-1.56 + 1.56i)T - 47iT^{2} \)
53 \( 1 + 1.55iT - 53T^{2} \)
59 \( 1 + (-4.00 + 4.00i)T - 59iT^{2} \)
61 \( 1 + 2.47T + 61T^{2} \)
67 \( 1 + (-5.37 + 5.37i)T - 67iT^{2} \)
71 \( 1 + (1.46 + 1.46i)T + 71iT^{2} \)
73 \( 1 + (5.34 + 5.34i)T + 73iT^{2} \)
79 \( 1 + 4.01T + 79T^{2} \)
83 \( 1 + (3.77 + 3.77i)T + 83iT^{2} \)
89 \( 1 + (4.28 - 4.28i)T - 89iT^{2} \)
97 \( 1 + (12.3 - 12.3i)T - 97iT^{2} \)
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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

−13.10730527894677912371212028119, −12.54763406665325950365119241555, −10.53750701438844024577161929528, −9.671483611915737630019249142145, −8.180349639925303447974388430020, −7.45355457223315784580283965963, −6.68052408427726847286842805010, −5.04050613182506521511301621614, −4.16987285180385074538004517521, −2.95492634407199452662433220458, 2.41377031387547969839910191874, 3.06608224131476190695808706070, 4.25652649925293724794321780986, 5.66224295409794872791014103046, 7.06661085516591694387067564466, 8.572591727068674593376778583589, 9.421200964010217417070986922289, 10.70763863339012886127381529833, 11.38990090106024353683879816700, 12.70437190812545083433206608330

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