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Properties

Label 2-14e2-7.2-c3-0-1
Degree $2$
Conductor $196$
Sign $0.968 - 0.250i$
Analytic cond. $11.5643$
Root an. cond. $3.40064$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5 − 8.66i)3-s + (−4 + 6.92i)5-s + (−36.5 + 63.2i)9-s + (20 + 34.6i)11-s + 12·13-s + 80·15-s + (−29 − 50.2i)17-s + (13 − 22.5i)19-s + (32 − 55.4i)23-s + (30.5 + 52.8i)25-s + 460.·27-s − 62·29-s + (126 + 218. i)31-s + (200. − 346. i)33-s + (−13 + 22.5i)37-s + ⋯
L(s)  = 1  + (−0.962 − 1.66i)3-s + (−0.357 + 0.619i)5-s + (−1.35 + 2.34i)9-s + (0.548 + 0.949i)11-s + 0.256·13-s + 1.37·15-s + (−0.413 − 0.716i)17-s + (0.156 − 0.271i)19-s + (0.290 − 0.502i)23-s + (0.244 + 0.422i)25-s + 3.27·27-s − 0.397·29-s + (0.730 + 1.26i)31-s + (1.05 − 1.82i)33-s + (−0.0577 + 0.100i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(11.5643\)
Root analytic conductor: \(3.40064\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :3/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.885121 + 0.112794i\)
\(L(\frac12)\) \(\approx\) \(0.885121 + 0.112794i\)
\(L(\frac{5}{2})\) 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 \( 1 + (5 + 8.66i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (4 - 6.92i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-20 - 34.6i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 12T + 2.19e3T^{2} \)
17 \( 1 + (29 + 50.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-13 + 22.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-32 + 55.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 62T + 2.43e4T^{2} \)
31 \( 1 + (-126 - 218. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (13 - 22.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 6T + 6.89e4T^{2} \)
43 \( 1 - 416T + 7.95e4T^{2} \)
47 \( 1 + (198 - 342. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-225 - 389. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-137 - 237. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (288 - 498. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-238 - 412. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 448T + 3.57e5T^{2} \)
73 \( 1 + (79 + 136. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-468 + 810. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 530T + 5.71e5T^{2} \)
89 \( 1 + (195 - 337. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 214T + 9.12e5T^{2} \)
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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

−12.07350063404896115632349366746, −11.36614372861511032803137633683, −10.53739689511351416743805860650, −8.885673835579649519320696782065, −7.47894079536926049896895715820, −7.03075473573254235327120519165, −6.11651032570314233543447536062, −4.77177480412931711420042403408, −2.63371530651747778750972825904, −1.16287013997763110956464220820, 0.52716305959210336679047639558, 3.56652895465981134140873328452, 4.36385182162356418492638610376, 5.50229500042108488130385230413, 6.35332264787187168497502308449, 8.354112142530244154607942557051, 9.165580909354311834219502877725, 10.10236776371355386289295420979, 11.13333769983969531706858604690, 11.60643708790950768327163042362

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