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Properties

Label 2-14e2-7.2-c5-0-15
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 $1$

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: $\chi_{196} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 196,\ (\ :5/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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

−11.27979147410590956352345878361, −9.819329012740139252359712869365, −9.057638799459447953799121919523, −7.42806055114733738623648555924, −7.02038825749366126230135094915, −5.65909378329434944995172405818, −4.62152546319849326830840778327, −2.73802194044855520860741891468, −1.29494279665711244268215306899, 0, 2.11230883532688811456742758286, 3.75420393136131650952869457377, 4.85861177277827643669995557948, 5.83712367443546048068760530086, 7.13267684656654766390511918258, 8.319713678401061769431551651672, 9.734044676077998866152408231377, 10.19367895286162410783903097434, 11.18169797244167781179143316440

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