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Properties

Label 2-14e2-196.111-c1-0-11
Degree $2$
Conductor $196$
Sign $0.998 + 0.0514i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
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  + (−0.349 + 1.37i)2-s + (−0.878 − 1.10i)3-s + (−1.75 − 0.958i)4-s + (−0.295 + 0.235i)5-s + (1.81 − 0.818i)6-s + (2.56 − 0.629i)7-s + (1.92 − 2.07i)8-s + (0.225 − 0.988i)9-s + (−0.219 − 0.487i)10-s + (−1.12 + 0.256i)11-s + (0.487 + 2.77i)12-s + (6.47 − 1.47i)13-s + (−0.0363 + 3.74i)14-s + (0.519 + 0.118i)15-s + (2.16 + 3.36i)16-s + (1.34 − 2.79i)17-s + ⋯
L(s)  = 1  + (−0.247 + 0.968i)2-s + (−0.507 − 0.636i)3-s + (−0.877 − 0.479i)4-s + (−0.132 + 0.105i)5-s + (0.741 − 0.334i)6-s + (0.971 − 0.237i)7-s + (0.681 − 0.732i)8-s + (0.0752 − 0.329i)9-s + (−0.0695 − 0.154i)10-s + (−0.339 + 0.0774i)11-s + (0.140 + 0.801i)12-s + (1.79 − 0.409i)13-s + (−0.00970 + 0.999i)14-s + (0.134 + 0.0306i)15-s + (0.541 + 0.840i)16-s + (0.326 − 0.678i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.998 + 0.0514i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ 0.998 + 0.0514i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.904991 - 0.0232968i\)
\(L(\frac12)\) \(\approx\) \(0.904991 - 0.0232968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.349 - 1.37i)T \)
7 \( 1 + (-2.56 + 0.629i)T \)
good3 \( 1 + (0.878 + 1.10i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (0.295 - 0.235i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (1.12 - 0.256i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-6.47 + 1.47i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-1.34 + 2.79i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 + 0.206T + 19T^{2} \)
23 \( 1 + (1.72 + 3.58i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (4.43 + 2.13i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 - 5.20T + 31T^{2} \)
37 \( 1 + (3.79 + 1.82i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-5.19 + 4.14i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (-0.460 - 0.367i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-1.70 - 7.49i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (10.5 - 5.07i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-1.76 + 2.21i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (0.733 - 1.52i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 - 7.21iT - 67T^{2} \)
71 \( 1 + (-6.50 - 13.5i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-2.24 - 0.512i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + 16.2iT - 79T^{2} \)
83 \( 1 + (2.90 - 12.7i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-0.958 - 0.218i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 - 14.0iT - 97T^{2} \)
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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

−12.67876335774577053540886885332, −11.42359471591503034226115364622, −10.62660648963237666650055045974, −9.247875747529926112655122391344, −8.160972833521362120548956497041, −7.40459758428317620065623668441, −6.29036745809875585882858587902, −5.42001953650794691067804805456, −3.97358524191237140592135785172, −1.11687364768077312540819572112, 1.72958408201761594059684883165, 3.72431497810314157553892803082, 4.73005090396295961684868659123, 5.84563842950524446747985447244, 7.936984445428663182831641268310, 8.560173835524927375387322875330, 9.823005326962437613246844739338, 10.84855576827192667173422680751, 11.21695803830401124738407182535, 12.17999017083527593141271113792

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