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Properties

Label 2-14e2-196.111-c1-0-13
Degree $2$
Conductor $196$
Sign $0.999 + 0.000891i$
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  + (−1.13 − 0.846i)2-s + (1.79 + 2.25i)3-s + (0.566 + 1.91i)4-s + (2.89 − 2.30i)5-s + (−0.127 − 4.07i)6-s + (−1.61 − 2.09i)7-s + (0.982 − 2.65i)8-s + (−1.18 + 5.17i)9-s + (−5.22 + 0.164i)10-s + (1.04 − 0.238i)11-s + (−3.30 + 4.72i)12-s + (1.70 − 0.388i)13-s + (0.0583 + 3.74i)14-s + (10.3 + 2.37i)15-s + (−3.35 + 2.17i)16-s + (−2.15 + 4.47i)17-s + ⋯
L(s)  = 1  + (−0.801 − 0.598i)2-s + (1.03 + 1.30i)3-s + (0.283 + 0.959i)4-s + (1.29 − 1.03i)5-s + (−0.0522 − 1.66i)6-s + (−0.611 − 0.791i)7-s + (0.347 − 0.937i)8-s + (−0.393 + 1.72i)9-s + (−1.65 + 0.0519i)10-s + (0.314 − 0.0718i)11-s + (−0.953 + 1.36i)12-s + (0.472 − 0.107i)13-s + (0.0155 + 0.999i)14-s + (2.68 + 0.612i)15-s + (−0.839 + 0.543i)16-s + (−0.522 + 1.08i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.000891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.999 + 0.000891i$
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: \((2,\ 196,\ (\ :1/2),\ 0.999 + 0.000891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26505 - 0.000563984i\)
\(L(\frac12)\) \(\approx\) \(1.26505 - 0.000563984i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 + 0.846i)T \)
7 \( 1 + (1.61 + 2.09i)T \)
good3 \( 1 + (-1.79 - 2.25i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-2.89 + 2.30i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (-1.04 + 0.238i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-1.70 + 0.388i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (2.15 - 4.47i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 + 5.54T + 19T^{2} \)
23 \( 1 + (-1.41 - 2.93i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (3.37 + 1.62i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + 3.97T + 31T^{2} \)
37 \( 1 + (-0.739 - 0.356i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-0.928 + 0.740i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (0.119 + 0.0951i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-1.04 - 4.58i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (9.27 - 4.46i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-4.35 + 5.45i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-5.04 + 10.4i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + 6.15iT - 67T^{2} \)
71 \( 1 + (2.60 + 5.40i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (12.6 + 2.88i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + 10.7iT - 79T^{2} \)
83 \( 1 + (2.65 - 11.6i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-14.5 - 3.31i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 - 4.59iT - 97T^{2} \)
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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

−12.84425157384000821077339349703, −10.90827079367995036458550659264, −10.29898613527894119434942439536, −9.375713334429760347249992690121, −9.030317475108074522962995252790, −8.051794342334286903266478232720, −6.25559048209597197431060345043, −4.47455198529290718746763205287, −3.52847065345882393897485417549, −1.89772162623190874095172712851, 1.94652721179471041450947332512, 2.73969694312499794617172353154, 5.82927688263784182177139260274, 6.63365334605281608335065596385, 7.16144675812605530305332986431, 8.664363078139701017471117013203, 9.144677231113018676432330809714, 10.17795088602520798724835935767, 11.40667781835469141118610820852, 12.91520063742106788963729753229

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