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Properties

Label 2-195-13.10-c1-0-5
Degree $2$
Conductor $195$
Sign $-0.964 + 0.265i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.36 + 1.36i)2-s + (−0.5 − 0.866i)3-s + (2.73 − 4.73i)4-s − i·5-s + (2.36 + 1.36i)6-s + (−2.13 − 1.23i)7-s + 9.46i·8-s + (−0.499 + 0.866i)9-s + (1.36 + 2.36i)10-s + (−3 + 1.73i)11-s − 5.46·12-s + (−0.866 + 3.5i)13-s + 6.73·14-s + (−0.866 + 0.5i)15-s + (−7.46 − 12.9i)16-s + (−1.63 + 2.83i)17-s + ⋯
L(s)  = 1  + (−1.67 + 0.965i)2-s + (−0.288 − 0.499i)3-s + (1.36 − 2.36i)4-s − 0.447i·5-s + (0.965 + 0.557i)6-s + (−0.806 − 0.465i)7-s + 3.34i·8-s + (−0.166 + 0.288i)9-s + (0.431 + 0.748i)10-s + (−0.904 + 0.522i)11-s − 1.57·12-s + (−0.240 + 0.970i)13-s + 1.79·14-s + (−0.223 + 0.129i)15-s + (−1.86 − 3.23i)16-s + (−0.396 + 0.686i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $-0.964 + 0.265i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (166, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 195,\ (\ :1/2),\ -0.964 + 0.265i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + iT \)
13 \( 1 + (0.866 - 3.5i)T \)
good2 \( 1 + (2.36 - 1.36i)T + (1 - 1.73i)T^{2} \)
7 \( 1 + (2.13 + 1.23i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.63 - 2.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.26 + 0.732i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.73 + 6.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.366 + 0.633i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.19iT - 31T^{2} \)
37 \( 1 + (3.46 - 2i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.56 - 4.36i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.86 - 3.23i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 + (9.29 + 5.36i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.23 + 2.13i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.79 + 2.76i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.02 - 4.63i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.39iT - 73T^{2} \)
79 \( 1 + 11.9T + 79T^{2} \)
83 \( 1 + 9.46iT - 83T^{2} \)
89 \( 1 + (-4.09 + 2.36i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.25 - 4.76i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.85705980556181283675128995646, −10.52902718505115553936630841451, −10.02671406873194932834810986308, −8.838104259991323430779224070010, −8.071107388898442515445064788810, −6.89859518039197401587681706611, −6.42296099324858962272383528015, −4.97054625871381084963059815421, −1.94952037092373564694330618721, 0, 2.54813575321205617070691676539, 3.50896798160505976905888056716, 5.82279468484964149591248070714, 7.25607357421259790594613944735, 8.218884784011416965996770463674, 9.330084619222913680730216101573, 10.02029253750458717383858462784, 10.74882316215381537301482318173, 11.58293530634186149756619759203

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