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Properties

Label 2-195-65.4-c1-0-9
Degree $2$
Conductor $195$
Sign $0.690 + 0.722i$
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 + (−3.08 − 1.12i)10-s + (−5.05 − 2.91i)11-s − 0.151i·12-s + (−3.21 − 1.63i)13-s + 6.35·14-s + (0.765 − 2.10i)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 + (−0.974 − 0.354i)10-s + (−1.52 − 0.879i)11-s − 0.0436i·12-s + (−0.890 − 0.454i)13-s + 1.69·14-s + (0.197 − 0.542i)15-s + (0.534 − 0.926i)16-s + (−0.604 + 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 + 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.690 + 0.722i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ 0.690 + 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67449 - 0.715986i\)
\(L(\frac12)\) \(\approx\) \(1.67449 - 0.715986i\)
\(L(\frac{3}{2})\) 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

−12.49443393961190409326120629920, −11.51248494476231221026563411140, −10.73501854011049435305001447938, −9.393464496823038698719093956291, −8.360929744125542384678934404133, −7.76450058802732739488238286179, −5.30754664235274343865054026663, −4.90132681069049337034607153478, −3.17164445771954997975517559344, −2.11528986324376228887676902141, 2.24278213801973036159013368586, 4.13340409774382272396749467484, 5.14325272100375052812073838999, 6.78702764106685651565831689894, 7.46863665158166706786454737324, 7.83458948384993343299136857078, 9.951920098845269317007256202061, 10.53917441403998813798535566318, 11.59957482246172121341982348473, 13.19073371255503826338688065166

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