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Properties

Label 2-14e2-7.5-c8-0-1
Degree $2$
Conductor $196$
Sign $-0.769 + 0.638i$
Analytic cond. $79.8462$
Root an. cond. $8.93567$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−60.8 + 35.1i)3-s + (978. + 564. i)5-s + (−807. + 1.39e3i)9-s + (4.40e3 + 7.63e3i)11-s − 1.37e4i·13-s − 7.94e4·15-s + (−4.46e4 + 2.57e4i)17-s + (−1.61e5 − 9.31e4i)19-s + (2.89e4 − 5.02e4i)23-s + (4.42e5 + 7.66e5i)25-s − 5.75e5i·27-s − 1.15e6·29-s + (−8.76e5 + 5.06e5i)31-s + (−5.36e5 − 3.09e5i)33-s + (8.73e5 − 1.51e6i)37-s + ⋯
L(s)  = 1  + (−0.751 + 0.434i)3-s + (1.56 + 0.903i)5-s + (−0.123 + 0.213i)9-s + (0.301 + 0.521i)11-s − 0.480i·13-s − 1.56·15-s + (−0.534 + 0.308i)17-s + (−1.23 − 0.714i)19-s + (0.103 − 0.179i)23-s + (1.13 + 1.96i)25-s − 1.08i·27-s − 1.63·29-s + (−0.949 + 0.548i)31-s + (−0.452 − 0.261i)33-s + (0.465 − 0.806i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.769 + 0.638i$
Analytic conductor: \(79.8462\)
Root analytic conductor: \(8.93567\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :4),\ -0.769 + 0.638i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.4855809630\)
\(L(\frac12)\) \(\approx\) \(0.4855809630\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (60.8 - 35.1i)T + (3.28e3 - 5.68e3i)T^{2} \)
5 \( 1 + (-978. - 564. i)T + (1.95e5 + 3.38e5i)T^{2} \)
11 \( 1 + (-4.40e3 - 7.63e3i)T + (-1.07e8 + 1.85e8i)T^{2} \)
13 \( 1 + 1.37e4iT - 8.15e8T^{2} \)
17 \( 1 + (4.46e4 - 2.57e4i)T + (3.48e9 - 6.04e9i)T^{2} \)
19 \( 1 + (1.61e5 + 9.31e4i)T + (8.49e9 + 1.47e10i)T^{2} \)
23 \( 1 + (-2.89e4 + 5.02e4i)T + (-3.91e10 - 6.78e10i)T^{2} \)
29 \( 1 + 1.15e6T + 5.00e11T^{2} \)
31 \( 1 + (8.76e5 - 5.06e5i)T + (4.26e11 - 7.38e11i)T^{2} \)
37 \( 1 + (-8.73e5 + 1.51e6i)T + (-1.75e12 - 3.04e12i)T^{2} \)
41 \( 1 - 5.16e6iT - 7.98e12T^{2} \)
43 \( 1 - 3.66e6T + 1.16e13T^{2} \)
47 \( 1 + (-4.92e6 - 2.84e6i)T + (1.19e13 + 2.06e13i)T^{2} \)
53 \( 1 + (3.25e5 + 5.63e5i)T + (-3.11e13 + 5.39e13i)T^{2} \)
59 \( 1 + (-5.14e6 + 2.96e6i)T + (7.34e13 - 1.27e14i)T^{2} \)
61 \( 1 + (2.11e6 + 1.22e6i)T + (9.58e13 + 1.66e14i)T^{2} \)
67 \( 1 + (1.42e7 + 2.45e7i)T + (-2.03e14 + 3.51e14i)T^{2} \)
71 \( 1 + 4.03e7T + 6.45e14T^{2} \)
73 \( 1 + (2.22e7 - 1.28e7i)T + (4.03e14 - 6.98e14i)T^{2} \)
79 \( 1 + (1.61e7 - 2.80e7i)T + (-7.58e14 - 1.31e15i)T^{2} \)
83 \( 1 + 7.88e7iT - 2.25e15T^{2} \)
89 \( 1 + (8.51e7 + 4.91e7i)T + (1.96e15 + 3.40e15i)T^{2} \)
97 \( 1 - 1.53e8iT - 7.83e15T^{2} \)
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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

−11.07791378433288970334377500445, −10.80281869299305719858271377760, −9.842304885393993513544155323559, −8.970740602584431158446540212612, −7.29721380053312717632982045903, −6.22868892751420619651189792855, −5.61477956006184486290061796496, −4.40914002388626714177387905021, −2.69627998968745248047622650647, −1.77002965231194363854574247456, 0.11520809871795932364111555504, 1.28841826839831701509544429990, 2.19614669303258787603873095117, 4.13341651645152785799557155525, 5.61903345318857405063528889692, 5.91994846881297657191273940164, 7.05278918479652362620726801157, 8.789725357754037864495257583535, 9.244318533749209399122401778717, 10.46462184058789626032858250691

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