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Properties

Label 2-14e2-28.27-c5-0-12
Degree $2$
Conductor $196$
Sign $-0.975 + 0.219i$
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  + (−0.565 + 5.62i)2-s + 16.9·3-s + (−31.3 − 6.36i)4-s − 9.31i·5-s + (−9.56 + 95.2i)6-s + (53.5 − 172. i)8-s + 43.5·9-s + (52.4 + 5.26i)10-s + 42.9i·11-s + (−530. − 107. i)12-s + 607. i·13-s − 157. i·15-s + (943. + 398. i)16-s + 568. i·17-s + (−24.5 + 244. i)18-s − 2.51e3·19-s + ⋯
L(s)  = 1  + (−0.0998 + 0.994i)2-s + 1.08·3-s + (−0.980 − 0.198i)4-s − 0.166i·5-s + (−0.108 + 1.08i)6-s + (0.295 − 0.955i)8-s + 0.179·9-s + (0.165 + 0.0166i)10-s + 0.107i·11-s + (−1.06 − 0.215i)12-s + 0.997i·13-s − 0.180i·15-s + (0.920 + 0.389i)16-s + 0.476i·17-s + (−0.0178 + 0.178i)18-s − 1.59·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.975 + 0.219i$
Analytic conductor: \(31.4352\)
Root analytic conductor: \(5.60671\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :5/2),\ -0.975 + 0.219i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.107031890\)
\(L(\frac12)\) \(\approx\) \(1.107031890\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.565 - 5.62i)T \)
7 \( 1 \)
good3 \( 1 - 16.9T + 243T^{2} \)
5 \( 1 + 9.31iT - 3.12e3T^{2} \)
11 \( 1 - 42.9iT - 1.61e5T^{2} \)
13 \( 1 - 607. iT - 3.71e5T^{2} \)
17 \( 1 - 568. iT - 1.41e6T^{2} \)
19 \( 1 + 2.51e3T + 2.47e6T^{2} \)
23 \( 1 - 2.05e3iT - 6.43e6T^{2} \)
29 \( 1 + 6.87e3T + 2.05e7T^{2} \)
31 \( 1 - 2.40e3T + 2.86e7T^{2} \)
37 \( 1 + 1.49e4T + 6.93e7T^{2} \)
41 \( 1 - 1.41e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.26e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.25e4T + 2.29e8T^{2} \)
53 \( 1 - 1.02e4T + 4.18e8T^{2} \)
59 \( 1 + 2.47e4T + 7.14e8T^{2} \)
61 \( 1 + 3.97e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.19e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.04e4iT - 1.80e9T^{2} \)
73 \( 1 + 3.84e4iT - 2.07e9T^{2} \)
79 \( 1 - 8.28e4iT - 3.07e9T^{2} \)
83 \( 1 + 5.23e4T + 3.93e9T^{2} \)
89 \( 1 - 384. iT - 5.58e9T^{2} \)
97 \( 1 + 4.95e4iT - 8.58e9T^{2} \)
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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

−12.53859855652110624632350258291, −10.99413798474761649856444018109, −9.660940785598896926109778813915, −8.892379337504356349862242181062, −8.231723579394378388248042347194, −7.15642659103070537404398454875, −6.08515433526271049856211668720, −4.64702201418334333865759383642, −3.54926655874802376504461004823, −1.79945909854449610517676098815, 0.28861165551966822362461551757, 2.07684002568769426750330969968, 2.99744075848361015432027199012, 4.05188376357100386401539675475, 5.51202424749800468776633527411, 7.31321306132347918169724185050, 8.560108494408197365317662176701, 8.892459543191374243489399494112, 10.26281641437967542518795866873, 10.85520847318506779723923110082

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