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Properties

Label 2-195-39.5-c1-0-19
Degree $2$
Conductor $195$
Sign $-0.606 - 0.795i$
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  + (−1.43 − 1.43i)2-s + (−0.158 − 1.72i)3-s + 2.13i·4-s + (−0.707 − 0.707i)5-s + (−2.25 + 2.70i)6-s + (−1.21 − 1.21i)7-s + (0.201 − 0.201i)8-s + (−2.94 + 0.546i)9-s + 2.03i·10-s + (−0.581 + 0.581i)11-s + (3.69 − 0.339i)12-s + (1.79 − 3.12i)13-s + 3.49i·14-s + (−1.10 + 1.33i)15-s + 3.70·16-s − 2.27·17-s + ⋯
L(s)  = 1  + (−1.01 − 1.01i)2-s + (−0.0915 − 0.995i)3-s + 1.06i·4-s + (−0.316 − 0.316i)5-s + (−0.919 + 1.10i)6-s + (−0.458 − 0.458i)7-s + (0.0710 − 0.0710i)8-s + (−0.983 + 0.182i)9-s + 0.643i·10-s + (−0.175 + 0.175i)11-s + (1.06 − 0.0979i)12-s + (0.496 − 0.868i)13-s + 0.933i·14-s + (−0.285 + 0.343i)15-s + 0.925·16-s − 0.552·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.606 - 0.795i$
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: \((2,\ 195,\ (\ :1/2),\ -0.606 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.161805 + 0.326867i\)
\(L(\frac12)\) \(\approx\) \(0.161805 + 0.326867i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.158 + 1.72i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-1.79 + 3.12i)T \)
good2 \( 1 + (1.43 + 1.43i)T + 2iT^{2} \)
7 \( 1 + (1.21 + 1.21i)T + 7iT^{2} \)
11 \( 1 + (0.581 - 0.581i)T - 11iT^{2} \)
17 \( 1 + 2.27T + 17T^{2} \)
19 \( 1 + (4.21 - 4.21i)T - 19iT^{2} \)
23 \( 1 - 3.13T + 23T^{2} \)
29 \( 1 + 3.12iT - 29T^{2} \)
31 \( 1 + (-2.36 + 2.36i)T - 31iT^{2} \)
37 \( 1 + (7.73 + 7.73i)T + 37iT^{2} \)
41 \( 1 + (5.66 + 5.66i)T + 41iT^{2} \)
43 \( 1 + 5.44iT - 43T^{2} \)
47 \( 1 + (3.80 - 3.80i)T - 47iT^{2} \)
53 \( 1 + 2.40iT - 53T^{2} \)
59 \( 1 + (-10.5 + 10.5i)T - 59iT^{2} \)
61 \( 1 - 5.20T + 61T^{2} \)
67 \( 1 + (7.77 - 7.77i)T - 67iT^{2} \)
71 \( 1 + (-5.74 - 5.74i)T + 71iT^{2} \)
73 \( 1 + (4.15 + 4.15i)T + 73iT^{2} \)
79 \( 1 + 9.23T + 79T^{2} \)
83 \( 1 + (-3.87 - 3.87i)T + 83iT^{2} \)
89 \( 1 + (4.74 - 4.74i)T - 89iT^{2} \)
97 \( 1 + (-9.51 + 9.51i)T - 97iT^{2} \)
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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

−11.79335784014789587842440643467, −10.85457489612557906339038093817, −10.10401671337580393403053716946, −8.746530802922495783151830963178, −8.161864054583795244606909356675, −6.99182765175052658114201094777, −5.64329239206736131494148710773, −3.55023375978474041488377151234, −2.04353890957573889150545915603, −0.43618267494876684700614933845, 3.18606193244094713909911639895, 4.75409603748328533071436047278, 6.21902975060054123238967267703, 6.91346975528004135090635891378, 8.540520894719544399377714677074, 8.869847865969329650268193588979, 9.937419314929480660297341496524, 10.86890140398433205964112135247, 11.84617499752503953681844574833, 13.28904611545874798893303101842

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