Properties

Label 2-14e2-7.4-c7-0-8
Degree $2$
Conductor $196$
Sign $0.968 + 0.250i$
Analytic cond. $61.2274$
Root an. cond. $7.82479$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−33.2 + 57.5i)3-s + (−78.6 − 136. i)5-s + (−1.11e3 − 1.92e3i)9-s + (−3.10e3 + 5.37e3i)11-s − 5.38e3·13-s + 1.04e4·15-s + (−5.49e3 + 9.52e3i)17-s + (5.85e3 + 1.01e4i)19-s + (−5.30e4 − 9.19e4i)23-s + (2.67e4 − 4.62e4i)25-s + 2.36e3·27-s − 5.15e4·29-s + (−1.23e5 + 2.14e5i)31-s + (−2.06e5 − 3.57e5i)33-s + (−2.16e5 − 3.75e5i)37-s + ⋯
L(s)  = 1  + (−0.709 + 1.22i)3-s + (−0.281 − 0.487i)5-s + (−0.508 − 0.880i)9-s + (−0.703 + 1.21i)11-s − 0.679·13-s + 0.798·15-s + (−0.271 + 0.470i)17-s + (0.195 + 0.338i)19-s + (−0.909 − 1.57i)23-s + (0.341 − 0.592i)25-s + 0.0231·27-s − 0.392·29-s + (−0.746 + 1.29i)31-s + (−0.998 − 1.72i)33-s + (−0.703 − 1.21i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(61.2274\)
Root analytic conductor: \(7.82479\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :7/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.5952016696\)
\(L(\frac12)\) \(\approx\) \(0.5952016696\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (33.2 - 57.5i)T + (-1.09e3 - 1.89e3i)T^{2} \)
5 \( 1 + (78.6 + 136. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (3.10e3 - 5.37e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + 5.38e3T + 6.27e7T^{2} \)
17 \( 1 + (5.49e3 - 9.52e3i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-5.85e3 - 1.01e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (5.30e4 + 9.19e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + 5.15e4T + 1.72e10T^{2} \)
31 \( 1 + (1.23e5 - 2.14e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (2.16e5 + 3.75e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 + 3.22e5T + 1.94e11T^{2} \)
43 \( 1 - 8.78e5T + 2.71e11T^{2} \)
47 \( 1 + (-3.27e5 - 5.67e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (-2.22e5 + 3.85e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-1.07e6 + 1.85e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-2.96e5 - 5.13e5i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (8.64e5 - 1.49e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 - 1.58e6T + 9.09e12T^{2} \)
73 \( 1 + (2.16e6 - 3.75e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (-3.04e6 - 5.26e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 8.10e6T + 2.71e13T^{2} \)
89 \( 1 + (-4.93e6 - 8.53e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 1.71e5T + 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.87014802673529094822249788769, −10.30650198671238385709322185531, −9.450721039593285566426778788209, −8.287661721969167721978780990657, −7.03503218981262130220260253290, −5.57364347261514177000946544966, −4.72504937983569954058748158577, −4.00333840680333740177532797361, −2.21423970157092315197588024821, −0.24913077485483568455998933811, 0.68671340316128552120876657741, 2.11830791649777830270543068506, 3.43063295030872095826499338234, 5.28528784141347941295761509725, 6.10365673643291093943405340781, 7.28985246771308818698677641825, 7.75736716040645705093470757519, 9.212435869940644546200925791535, 10.59611569675871439929351689078, 11.49184141311184676169379759121

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