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Properties

Label 2-14e2-7.2-c5-0-1
Degree $2$
Conductor $196$
Sign $-0.968 + 0.250i$
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  + (13 + 22.5i)3-s + (8 − 13.8i)5-s + (−216. + 374. i)9-s + (−4 − 6.92i)11-s − 684·13-s + 415.·15-s + (−1.10e3 − 1.92e3i)17-s + (−1.34e3 + 2.33e3i)19-s + (−1.67e3 + 2.89e3i)23-s + (1.43e3 + 2.48e3i)25-s − 4.93e3·27-s − 3.25e3·29-s + (2.39e3 + 4.14e3i)31-s + (103. − 180. i)33-s + (5.73e3 − 9.93e3i)37-s + ⋯
L(s)  = 1  + (0.833 + 1.44i)3-s + (0.143 − 0.247i)5-s + (−0.890 + 1.54i)9-s + (−0.00996 − 0.0172i)11-s − 1.12·13-s + 0.477·15-s + (−0.930 − 1.61i)17-s + (−0.857 + 1.48i)19-s + (−0.659 + 1.14i)23-s + (0.459 + 0.795i)25-s − 1.30·27-s − 0.718·29-s + (0.447 + 0.774i)31-s + (0.0166 − 0.0287i)33-s + (0.688 − 1.19i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(31.4352\)
Root analytic conductor: \(5.60671\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :5/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.155462419\)
\(L(\frac12)\) \(\approx\) \(1.155462419\)
\(L(\frac{7}{2})\) 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 \( 1 + (-13 - 22.5i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-8 + 13.8i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (4 + 6.92i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 684T + 3.71e5T^{2} \)
17 \( 1 + (1.10e3 + 1.92e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.34e3 - 2.33e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.67e3 - 2.89e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 3.25e3T + 2.05e7T^{2} \)
31 \( 1 + (-2.39e3 - 4.14e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.73e3 + 9.93e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.33e4T + 1.15e8T^{2} \)
43 \( 1 + 928T + 1.47e8T^{2} \)
47 \( 1 + (-606 + 1.04e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (6.55e3 + 1.13e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.73e4 - 3.00e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (516 - 893. i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (5.05e3 + 8.75e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 6.27e4T + 1.80e9T^{2} \)
73 \( 1 + (9.46e3 + 1.63e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (5.70e3 - 9.87e3i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 8.89e4T + 3.93e9T^{2} \)
89 \( 1 + (-9.86e3 + 1.70e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.70e4T + 8.58e9T^{2} \)
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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

−12.02966958913198066133698282830, −10.92673191329357182765768426517, −9.879066792636774397379323908682, −9.386824691810031354771058252742, −8.415155578098140350819834503166, −7.24750079095048590274414887969, −5.45127001446843719511733624744, −4.54997718259751957701985424674, −3.45581213551854127201192818844, −2.18935392985078969647055405330, 0.28414091964357861175895426635, 1.97121245940668773411919860207, 2.67725460591351484186814211517, 4.43222895904413157418818158283, 6.34703994555528309458923664387, 6.86738153233837824018392239054, 8.102083328796753104697967181473, 8.692491983641107834103665548454, 10.01897694365593303836604647588, 11.23152681717535060815148447484

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