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Properties

Label 2-195-65.49-c1-0-1
Degree $2$
Conductor $195$
Sign $0.402 - 0.915i$
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  + (−0.733 − 1.27i)2-s + (−0.866 + 0.5i)3-s + (−0.0756 + 0.131i)4-s + (0.387 + 2.20i)5-s + (1.27 + 0.733i)6-s + (−2.16 + 3.75i)7-s − 2.71·8-s + (0.499 − 0.866i)9-s + (2.51 − 2.10i)10-s + (−5.05 + 2.91i)11-s − 0.151i·12-s + (3.21 − 1.63i)13-s + 6.35·14-s + (−1.43 − 1.71i)15-s + (2.13 + 3.70i)16-s + (2.49 + 1.43i)17-s + ⋯
L(s)  = 1  + (−0.518 − 0.898i)2-s + (−0.499 + 0.288i)3-s + (−0.0378 + 0.0655i)4-s + (0.173 + 0.984i)5-s + (0.518 + 0.299i)6-s + (−0.818 + 1.41i)7-s − 0.958·8-s + (0.166 − 0.288i)9-s + (0.794 − 0.666i)10-s + (−1.52 + 0.879i)11-s − 0.0436i·12-s + (0.890 − 0.454i)13-s + 1.69·14-s + (−0.371 − 0.442i)15-s + (0.534 + 0.926i)16-s + (0.604 + 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.402 - 0.915i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.402 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.455505 + 0.297361i\)
\(L(\frac12)\) \(\approx\) \(0.455505 + 0.297361i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.387 - 2.20i)T \)
13 \( 1 + (-3.21 + 1.63i)T \)
good2 \( 1 + (0.733 + 1.27i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (2.16 - 3.75i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.05 - 2.91i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.49 - 1.43i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.89 - 1.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.07 - 0.623i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.86 + 4.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.880iT - 31T^{2} \)
37 \( 1 + (0.0960 + 0.166i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.198 + 0.114i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.27 - 2.47i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.0904T + 47T^{2} \)
53 \( 1 - 4.46iT - 53T^{2} \)
59 \( 1 + (-6.48 - 3.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.78 - 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.73 + 6.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.08 - 1.77i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.62T + 73T^{2} \)
79 \( 1 - 6.75T + 79T^{2} \)
83 \( 1 - 9.36T + 83T^{2} \)
89 \( 1 + (-13.1 + 7.60i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.762 - 1.32i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.34820842371619395046142579725, −11.59064796426256485595922474114, −10.52180537953813051707760317586, −10.05830122750818895657575943144, −9.189724355267662114762451876867, −7.72630929271202913420354228843, −6.08999302011479356422995452557, −5.63026548791387137575137627859, −3.28792502323226629514523950607, −2.31566199885786404865553607545, 0.57773537593112699213721366318, 3.44823390838902557085771955898, 5.21103457286531828544383780540, 6.20606640743718196685719715835, 7.29660163628844862249682665004, 8.052537555784886301297112509695, 9.141137145847199702501186251814, 10.24382690341797220355246425567, 11.28234022062899841362705338796, 12.54878765666637425158840173151

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