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Properties

Label 2-195-13.4-c1-0-9
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  + (−0.633 − 0.366i)2-s + (−0.5 + 0.866i)3-s + (−0.732 − 1.26i)4-s − i·5-s + (0.633 − 0.366i)6-s + (−3.86 + 2.23i)7-s + 2.53i·8-s + (−0.499 − 0.866i)9-s + (−0.366 + 0.633i)10-s + (−3 − 1.73i)11-s + 1.46·12-s + (0.866 + 3.5i)13-s + 3.26·14-s + (0.866 + 0.5i)15-s + (−0.535 + 0.928i)16-s + (−3.36 − 5.83i)17-s + ⋯
L(s)  = 1  + (−0.448 − 0.258i)2-s + (−0.288 + 0.499i)3-s + (−0.366 − 0.633i)4-s − 0.447i·5-s + (0.258 − 0.149i)6-s + (−1.46 + 0.843i)7-s + 0.896i·8-s + (−0.166 − 0.288i)9-s + (−0.115 + 0.200i)10-s + (−0.904 − 0.522i)11-s + 0.422·12-s + (0.240 + 0.970i)13-s + 0.873·14-s + (0.223 + 0.129i)15-s + (−0.133 + 0.232i)16-s + (−0.816 − 1.41i)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: $\chi_{195} (121, \cdot )$
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 \( 1 + (0.633 + 0.366i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (3.86 - 2.23i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.36 + 5.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.73 - 2.73i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.267 - 0.464i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.36 + 2.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.19iT - 31T^{2} \)
37 \( 1 + (-3.46 - 2i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.56 - 2.63i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.133 + 0.232i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 0.196iT - 47T^{2} \)
53 \( 1 + 6.92T + 53T^{2} \)
59 \( 1 + (-6.29 + 3.63i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.23 + 3.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.7 + 6.23i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.0 - 6.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 - 1.92T + 79T^{2} \)
83 \( 1 + 2.53iT - 83T^{2} \)
89 \( 1 + (1.09 + 0.633i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (14.2 - 8.23i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.78486571897496640556099504322, −10.88704246080825900499531077833, −9.747805187254568724717075269251, −9.297837271694347772817179524443, −8.397545943136557983225476792160, −6.44983819931709928717283232723, −5.63274898648687152843210373417, −4.41672072458009683385055301453, −2.57984723261465127529419658872, 0, 2.93826811412782528210646961304, 4.22863212442514279364144663292, 6.16768119854139933170357530447, 6.97918133874968014150286173064, 7.84687894342443839980628648288, 8.940008877377755368924514345575, 10.27746498147660912559917425332, 10.69273146676905111689614018178, 12.52818364570314344649662220585

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