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Properties

Label 2-14e2-7.4-c5-0-13
Degree $2$
Conductor $196$
Sign $-0.386 + 0.922i$
Analytic cond. $31.4352$
Root an. cond. $5.60671$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8 − 13.8i)3-s + (−8 − 13.8i)5-s + (−6.49 − 11.2i)9-s + (38 − 65.8i)11-s + 880·13-s − 255.·15-s + (528 − 914. i)17-s + (−968 − 1.67e3i)19-s + (−468 − 810. i)23-s + (1.43e3 − 2.48e3i)25-s + 3.68e3·27-s − 3.98e3·29-s + (−784 + 1.35e3i)31-s + (−607. − 1.05e3i)33-s + (−2.46e3 − 4.27e3i)37-s + ⋯
L(s)  = 1  + (0.513 − 0.888i)3-s + (−0.143 − 0.247i)5-s + (−0.0267 − 0.0463i)9-s + (0.0946 − 0.164i)11-s + 1.44·13-s − 0.293·15-s + (0.443 − 0.767i)17-s + (−0.615 − 1.06i)19-s + (−0.184 − 0.319i)23-s + (0.459 − 0.795i)25-s + 0.971·27-s − 0.879·29-s + (−0.146 + 0.253i)31-s + (−0.0971 − 0.168i)33-s + (−0.296 − 0.513i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(31.4352\)
Root analytic conductor: \(5.60671\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :5/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.223325416\)
\(L(\frac12)\) \(\approx\) \(2.223325416\)
\(L(\frac{7}{2})\) 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 \( 1 + (-8 + 13.8i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (8 + 13.8i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-38 + 65.8i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 880T + 3.71e5T^{2} \)
17 \( 1 + (-528 + 914. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (968 + 1.67e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (468 + 810. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 3.98e3T + 2.05e7T^{2} \)
31 \( 1 + (784 - 1.35e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.46e3 + 4.27e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.58e4T + 1.15e8T^{2} \)
43 \( 1 + 1.64e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.03e4 - 1.79e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.87e4 + 3.23e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.05e4 - 1.83e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.49e3 - 2.59e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-2.29e4 + 3.96e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 4.98e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.81e4 + 4.87e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (2.03e4 + 3.52e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 1.12e5T + 3.93e9T^{2} \)
89 \( 1 + (3.21e4 + 5.56e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 2.27e3T + 8.58e9T^{2} \)
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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

−11.40397241812851182311984451600, −10.41391690129658148311588028599, −8.942898202073100229609197807777, −8.337777881192578572905154500694, −7.23631401458945937679305316037, −6.30201142062489954899870952425, −4.83078896996562077526495690341, −3.34176579791895165297360408445, −1.95317081922379601423345697329, −0.66966273850264665563748043896, 1.53492376867214420360106214468, 3.43351682167006167427289163902, 3.97068208298486547358754977853, 5.55214891417132274776228048930, 6.73325904190581035123164386360, 8.169400650125239181379762570001, 8.913900183961893331339833135641, 10.03886602568073940224929296641, 10.69561159184162142957522118339, 11.81561721452964933516469377766

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