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Properties

Label 2-14e2-28.27-c1-0-15
Degree $2$
Conductor $196$
Sign $-0.654 + 0.755i$
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  + (1 − i)2-s − 1.73·3-s − 2i·4-s − 1.73i·5-s + (−1.73 + 1.73i)6-s + (−2 − 2i)8-s + (−1.73 − 1.73i)10-s − i·11-s + 3.46i·12-s − 3.46i·13-s + 2.99i·15-s − 4·16-s + 1.73i·17-s + 5.19·19-s − 3.46·20-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00·3-s − i·4-s − 0.774i·5-s + (−0.707 + 0.707i)6-s + (−0.707 − 0.707i)8-s + (−0.547 − 0.547i)10-s − 0.301i·11-s + 0.999i·12-s − 0.960i·13-s + 0.774i·15-s − 16-s + 0.420i·17-s + 1.19·19-s − 0.774·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.654 + 0.755i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1/2),\ -0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465468 - 1.01886i\)
\(L(\frac12)\) \(\approx\) \(0.465468 - 1.01886i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
7 \( 1 \)
good3 \( 1 + 1.73T + 3T^{2} \)
5 \( 1 + 1.73iT - 5T^{2} \)
11 \( 1 + iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
19 \( 1 - 5.19T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 1.73T + 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 8.66T + 47T^{2} \)
53 \( 1 + T + 53T^{2} \)
59 \( 1 - 5.19T + 59T^{2} \)
61 \( 1 + 5.19iT - 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 + 14iT - 71T^{2} \)
73 \( 1 + 8.66iT - 73T^{2} \)
79 \( 1 - 9iT - 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 15.5iT - 89T^{2} \)
97 \( 1 - 17.3iT - 97T^{2} \)
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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

−12.14174549877362005659332399937, −11.35474996207384170949455906776, −10.51507044874945499850916154571, −9.492291904404912654301797641208, −8.188961128848437569121324935635, −6.44514655592116981277238489987, −5.48057980755095984970373669333, −4.78254900792048526307812490851, −3.15165488177483442214732883951, −0.947965969729642590573301671816, 2.92764804111139807807848556884, 4.50936342399112550006442432701, 5.56360620404153486167880400302, 6.60595923383766732206431824027, 7.24754670993741653361584670891, 8.679538626874350344124581681602, 10.05526867009601357100975205999, 11.40041020560961299174732285684, 11.72406216880972544597594119078, 12.82908247304820185973761795621

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