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Properties

Label 2-195-65.29-c1-0-8
Degree $2$
Conductor $195$
Sign $-0.0562 + 0.998i$
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  + (−1.16 + 0.672i)2-s + (0.866 − 0.5i)3-s + (−0.0962 + 0.166i)4-s + (−0.868 − 2.06i)5-s + (−0.672 + 1.16i)6-s + (−3.39 − 1.96i)7-s − 2.94i·8-s + (0.499 − 0.866i)9-s + (2.39 + 1.81i)10-s + (−1.37 − 2.38i)11-s + 0.192i·12-s + (1.14 + 3.41i)13-s + 5.27·14-s + (−1.78 − 1.35i)15-s + (1.78 + 3.09i)16-s + (−4.09 − 2.36i)17-s + ⋯
L(s)  = 1  + (−0.823 + 0.475i)2-s + (0.499 − 0.288i)3-s + (−0.0481 + 0.0833i)4-s + (−0.388 − 0.921i)5-s + (−0.274 + 0.475i)6-s + (−1.28 − 0.741i)7-s − 1.04i·8-s + (0.166 − 0.288i)9-s + (0.757 + 0.574i)10-s + (−0.414 − 0.718i)11-s + 0.0555i·12-s + (0.318 + 0.947i)13-s + 1.40·14-s + (−0.460 − 0.348i)15-s + (0.447 + 0.774i)16-s + (−0.992 − 0.573i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0562 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0562 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.0562 + 0.998i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.0562 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.357242 - 0.377920i\)
\(L(\frac12)\) \(\approx\) \(0.357242 - 0.377920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (0.868 + 2.06i)T \)
13 \( 1 + (-1.14 - 3.41i)T \)
good2 \( 1 + (1.16 - 0.672i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (3.39 + 1.96i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.37 + 2.38i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (4.09 + 2.36i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.85 + 3.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.57 + 2.64i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.31 + 2.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.71T + 31T^{2} \)
37 \( 1 + (3.66 - 2.11i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.408 + 0.708i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.85 + 2.80i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.36iT - 47T^{2} \)
53 \( 1 - 7.01iT - 53T^{2} \)
59 \( 1 + (-1.09 + 1.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.41 + 11.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.8 + 6.28i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.08 - 10.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.955iT - 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 + 11.7iT - 83T^{2} \)
89 \( 1 + (5.60 + 9.70i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.52 - 0.882i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.51946756268296751599721750388, −11.21983512794192832427818687188, −9.775038257468935000099878812501, −9.067104557080566738893625853805, −8.394825027883290144969688384421, −7.22188381172126208689062288399, −6.52255167208677345393788536783, −4.48454816943196612805358250569, −3.27280263554655665597698558789, −0.55601591134372578737151110094, 2.41320646052629875953579623042, 3.47948158888854427688343089521, 5.40081948652558687486067958046, 6.72802911358997988751986074421, 8.022330744053985414683523784148, 8.962920223161357167781912217064, 9.969908313904892954348735549927, 10.40929921667142518422027357922, 11.50483157188593163382011329484, 12.70256121483893138325532827281

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