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Properties

Label 2-14e2-4.3-c2-0-17
Degree $2$
Conductor $196$
Sign $0.986 + 0.163i$
Analytic cond. $5.34061$
Root an. cond. $2.31097$
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  + (−0.163 + 1.99i)2-s − 1.56i·3-s + (−3.94 − 0.652i)4-s − 3.43·5-s + (3.11 + 0.255i)6-s + (1.94 − 7.75i)8-s + 6.56·9-s + (0.562 − 6.85i)10-s − 8.48i·11-s + (−1.01 + 6.15i)12-s + 18.5·13-s + 5.36i·15-s + (15.1 + 5.14i)16-s + 8.87·17-s + (−1.07 + 13.0i)18-s − 30.3i·19-s + ⋯
L(s)  = 1  + (−0.0818 + 0.996i)2-s − 0.520i·3-s + (−0.986 − 0.163i)4-s − 0.687·5-s + (0.518 + 0.0425i)6-s + (0.243 − 0.969i)8-s + 0.729·9-s + (0.0562 − 0.685i)10-s − 0.771i·11-s + (−0.0848 + 0.513i)12-s + 1.42·13-s + 0.357i·15-s + (0.946 + 0.321i)16-s + 0.522·17-s + (−0.0596 + 0.727i)18-s − 1.59i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.986 + 0.163i$
Analytic conductor: \(5.34061\)
Root analytic conductor: \(2.31097\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1),\ 0.986 + 0.163i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19932 - 0.0984728i\)
\(L(\frac12)\) \(\approx\) \(1.19932 - 0.0984728i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.163 - 1.99i)T \)
7 \( 1 \)
good3 \( 1 + 1.56iT - 9T^{2} \)
5 \( 1 + 3.43T + 25T^{2} \)
11 \( 1 + 8.48iT - 121T^{2} \)
13 \( 1 - 18.5T + 169T^{2} \)
17 \( 1 - 8.87T + 289T^{2} \)
19 \( 1 + 30.3iT - 361T^{2} \)
23 \( 1 + 26.5iT - 529T^{2} \)
29 \( 1 - 18.6T + 841T^{2} \)
31 \( 1 - 41.2iT - 961T^{2} \)
37 \( 1 + 3.49T + 1.36e3T^{2} \)
41 \( 1 + 37.7T + 1.68e3T^{2} \)
43 \( 1 - 50.8iT - 1.84e3T^{2} \)
47 \( 1 + 51.9iT - 2.20e3T^{2} \)
53 \( 1 - 15.3T + 2.80e3T^{2} \)
59 \( 1 + 38.3iT - 3.48e3T^{2} \)
61 \( 1 - 72.5T + 3.72e3T^{2} \)
67 \( 1 + 32.0iT - 4.48e3T^{2} \)
71 \( 1 + 50.6iT - 5.04e3T^{2} \)
73 \( 1 - 5.48T + 5.32e3T^{2} \)
79 \( 1 - 39.8iT - 6.24e3T^{2} \)
83 \( 1 + 4.28iT - 6.88e3T^{2} \)
89 \( 1 + 123.T + 7.92e3T^{2} \)
97 \( 1 - 32.0T + 9.40e3T^{2} \)
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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

−12.50811315504664206225721481530, −11.29501083228401838353436098365, −10.18460835640166844495329291009, −8.762168633840960997586651954194, −8.178910610783806967921860055223, −7.01749126911249557734378196498, −6.29262473898675275503437899695, −4.82782975723588471917720236995, −3.57291456648811640900141381122, −0.847200470797146939971092553335, 1.52687137800610861333612486973, 3.61429495229511904029602709851, 4.17438293641876742573216538538, 5.65386215857781481928907024517, 7.50762031557753137184078699722, 8.435553249239355992626134863413, 9.698674694845110983696981370085, 10.26433882110200072452988938863, 11.34703516319960888671276738891, 12.11716565652157165654678914894

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