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Properties

Label 2-195-195.8-c2-0-21
Degree $2$
Conductor $195$
Sign $0.855 + 0.517i$
Analytic cond. $5.31336$
Root an. cond. $2.30507$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82i·2-s + (1.29 + 2.70i)3-s − 4.00·4-s + (4.94 + 0.707i)5-s + (7.65 − 3.65i)6-s + 5i·7-s + (−5.65 + 7i)9-s + (2.00 − 14.0i)10-s + (9.19 + 9.19i)11-s + (−5.17 − 10.8i)12-s + 13i·13-s + 14.1·14-s + (4.48 + 14.3i)15-s − 15.9·16-s + (13.4 − 13.4i)17-s + (19.7 + 16i)18-s + ⋯
L(s)  = 1  − 1.41i·2-s + (0.430 + 0.902i)3-s − 1.00·4-s + (0.989 + 0.141i)5-s + (1.27 − 0.609i)6-s + 0.714i·7-s + (−0.628 + 0.777i)9-s + (0.200 − 1.40i)10-s + (0.835 + 0.835i)11-s + (−0.430 − 0.902i)12-s + i·13-s + 1.01·14-s + (0.299 + 0.954i)15-s − 0.999·16-s + (0.790 − 0.790i)17-s + (1.09 + 0.888i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(195\)    =    \(3 \cdot 5 \cdot 13\)
Sign: $0.855 + 0.517i$
Analytic conductor: \(5.31336\)
Root analytic conductor: \(2.30507\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{195} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 195,\ (\ :1),\ 0.855 + 0.517i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.95557 - 0.545339i\)
\(L(\frac12)\) \(\approx\) \(1.95557 - 0.545339i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.29 - 2.70i)T \)
5 \( 1 + (-4.94 - 0.707i)T \)
13 \( 1 - 13iT \)
good2 \( 1 + 2.82iT - 4T^{2} \)
7 \( 1 - 5iT - 49T^{2} \)
11 \( 1 + (-9.19 - 9.19i)T + 121iT^{2} \)
17 \( 1 + (-13.4 + 13.4i)T - 289iT^{2} \)
19 \( 1 + (-22 + 22i)T - 361iT^{2} \)
23 \( 1 + (14.8 + 14.8i)T + 529iT^{2} \)
29 \( 1 + 38.1T + 841T^{2} \)
31 \( 1 + (13 + 13i)T + 961iT^{2} \)
37 \( 1 - 15iT - 1.36e3T^{2} \)
41 \( 1 + (-24.7 + 24.7i)T - 1.68e3iT^{2} \)
43 \( 1 + (-17 + 17i)T - 1.84e3iT^{2} \)
47 \( 1 + 72.1T + 2.20e3T^{2} \)
53 \( 1 + (9.19 - 9.19i)T - 2.80e3iT^{2} \)
59 \( 1 + (-48.0 + 48.0i)T - 3.48e3iT^{2} \)
61 \( 1 - 67T + 3.72e3T^{2} \)
67 \( 1 + 100T + 4.48e3T^{2} \)
71 \( 1 + (37.4 - 37.4i)T - 5.04e3iT^{2} \)
73 \( 1 + 76T + 5.32e3T^{2} \)
79 \( 1 - 89iT - 6.24e3T^{2} \)
83 \( 1 + 93.3T + 6.88e3T^{2} \)
89 \( 1 + (4.94 - 4.94i)T - 7.92e3iT^{2} \)
97 \( 1 + 25T + 9.40e3T^{2} \)
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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

−11.85813531206411174550288209091, −11.30050843573170060686677211526, −10.04850133208157985966521014489, −9.448754754343788795217759755295, −9.039614040141250536433508342698, −7.02380470192875419509232447698, −5.42109723468143709036723745312, −4.23103267147457906194518557479, −2.87654534543550957148819516136, −1.88834231531864293736918129943, 1.38502037939227005404942432402, 3.48536693513666846467911814275, 5.70342643038456322478247373993, 5.99125190313169500402224879531, 7.31291958751473368567482752119, 7.972136456950412413237786555144, 8.983778033987441202315107925259, 10.07024208328800840404546263441, 11.53353261396605113628476495971, 12.84775981696569218941013065500

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