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Properties

Label 2-14e2-28.11-c2-0-35
Degree $2$
Conductor $196$
Sign $-0.0100 - 0.999i$
Analytic cond. $5.34061$
Root an. cond. $2.31097$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.259 − 1.98i)2-s + (−2.58 − 1.49i)3-s + (−3.86 − 1.02i)4-s + (−3.40 − 5.89i)5-s + (−3.62 + 4.73i)6-s + (−3.04 + 7.39i)8-s + (−0.0457 − 0.0792i)9-s + (−12.5 + 5.21i)10-s + (12.4 + 7.17i)11-s + (8.45 + 8.42i)12-s − 4.62·13-s + 20.3i·15-s + (13.8 + 7.95i)16-s + (5.80 − 10.0i)17-s + (−0.168 + 0.0701i)18-s + (−19.4 + 11.2i)19-s + ⋯
L(s)  = 1  + (0.129 − 0.991i)2-s + (−0.861 − 0.497i)3-s + (−0.966 − 0.257i)4-s + (−0.680 − 1.17i)5-s + (−0.604 + 0.789i)6-s + (−0.380 + 0.924i)8-s + (−0.00508 − 0.00880i)9-s + (−1.25 + 0.521i)10-s + (1.12 + 0.651i)11-s + (0.704 + 0.702i)12-s − 0.355·13-s + 1.35i·15-s + (0.867 + 0.496i)16-s + (0.341 − 0.591i)17-s + (−0.00938 + 0.00389i)18-s + (−1.02 + 0.590i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0100 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0100 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(196\)    =    \(2^{2} \cdot 7^{2}\)
Sign: $-0.0100 - 0.999i$
Analytic conductor: \(5.34061\)
Root analytic conductor: \(2.31097\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{196} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 196,\ (\ :1),\ -0.0100 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.232539 + 0.234887i\)
\(L(\frac12)\) \(\approx\) \(0.232539 + 0.234887i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.259 + 1.98i)T \)
7 \( 1 \)
good3 \( 1 + (2.58 + 1.49i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (3.40 + 5.89i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (-12.4 - 7.17i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 4.62T + 169T^{2} \)
17 \( 1 + (-5.80 + 10.0i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (19.4 - 11.2i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-0.739 + 0.426i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 42.8T + 841T^{2} \)
31 \( 1 + (1.34 + 0.774i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (22.5 + 38.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 36.6T + 1.68e3T^{2} \)
43 \( 1 + 27.9iT - 1.84e3T^{2} \)
47 \( 1 + (36.5 - 21.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (21.7 - 37.5i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (46.7 + 26.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (7.97 + 13.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (79.6 + 45.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 16.3iT - 5.04e3T^{2} \)
73 \( 1 + (-4.79 + 8.30i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (49.3 - 28.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 14.3iT - 6.88e3T^{2} \)
89 \( 1 + (-50.0 - 86.7i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 68.2T + 9.40e3T^{2} \)
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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

−11.77959108842396672148521581380, −10.95146222085559687020292284728, −9.521184212288579425143253002759, −8.825369109446719596373507262666, −7.48224360158096012655426044940, −5.97259185711137205346496247939, −4.82417075878586233468904039335, −3.83736759478703208251734075247, −1.58069092327661275841665264058, −0.21292756149006271899920101026, 3.45723138965212881566370065257, 4.53075554123609177020578434419, 5.90973488745466191120722845875, 6.64626871598520277448700311498, 7.70684979228507735516033839031, 8.872303901448528283575719403544, 10.15733490721954642872599819951, 11.09898825911558775283091360452, 11.80788228763925629217902212982, 13.10664934396372708436480992861

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