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Properties

Label 2-13e2-13.11-c2-0-12
Degree $2$
Conductor $169$
Sign $0.852 + 0.522i$
Analytic cond. $4.60491$
Root an. cond. $2.14590$
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  + (0.133 − 0.5i)2-s + (0.366 + 0.633i)3-s + (3.23 + 1.86i)4-s + (−2.63 − 2.63i)5-s + (0.366 − 0.0980i)6-s + (−1.53 − 5.73i)7-s + (2.83 − 2.83i)8-s + (4.23 − 7.33i)9-s + (−1.66 + 0.964i)10-s + (15.6 + 4.19i)11-s + 2.73i·12-s − 3.07·14-s + (0.705 − 2.63i)15-s + (6.42 + 11.1i)16-s + (15.9 + 9.23i)17-s + (−3.09 − 3.09i)18-s + ⋯
L(s)  = 1  + (0.0669 − 0.250i)2-s + (0.122 + 0.211i)3-s + (0.808 + 0.466i)4-s + (−0.526 − 0.526i)5-s + (0.0610 − 0.0163i)6-s + (−0.219 − 0.818i)7-s + (0.353 − 0.353i)8-s + (0.470 − 0.814i)9-s + (−0.166 + 0.0964i)10-s + (1.42 + 0.381i)11-s + 0.227i·12-s − 0.219·14-s + (0.0470 − 0.175i)15-s + (0.401 + 0.695i)16-s + (0.940 + 0.543i)17-s + (−0.172 − 0.172i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.852 + 0.522i$
Analytic conductor: \(4.60491\)
Root analytic conductor: \(2.14590\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1),\ 0.852 + 0.522i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.79179 - 0.504962i\)
\(L(\frac12)\) \(\approx\) \(1.79179 - 0.504962i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-0.133 + 0.5i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-0.366 - 0.633i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (2.63 + 2.63i)T + 25iT^{2} \)
7 \( 1 + (1.53 + 5.73i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-15.6 - 4.19i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (-15.9 - 9.23i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (6.09 - 1.63i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (17.4 - 10.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (4.69 + 8.13i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (11.9 + 11.9i)T + 961iT^{2} \)
37 \( 1 + (30.2 + 8.11i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (12.0 - 44.9i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (45 + 25.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (34.3 - 34.3i)T - 2.20e3iT^{2} \)
53 \( 1 + 14.7T + 2.80e3T^{2} \)
59 \( 1 + (-24.9 - 92.9i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (12.8 - 22.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (10.4 - 39.0i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-44.6 + 11.9i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (19.2 - 19.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 62.7T + 6.24e3T^{2} \)
83 \( 1 + (24.4 + 24.4i)T + 6.88e3iT^{2} \)
89 \( 1 + (-86.4 - 23.1i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (52.9 - 14.1i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−12.15139942959375358410048998769, −11.83517893371774597570948673026, −10.45032893668560166590433737725, −9.585617661028197993012217431031, −8.271698851041554459822756419829, −7.19930402568425016979666822779, −6.30716913103000176746117846904, −4.14791373739191080541392047294, −3.62405518700637753779762700225, −1.36811308369399540351956679160, 1.82730030299833389328326298374, 3.37791415516011469944429060809, 5.21438274372553829225241008665, 6.44237724147693096475645598753, 7.22449004355914587931243235774, 8.353753322412079474346583767151, 9.688404186504000311628336794029, 10.78848824993885765758217165106, 11.66283370122462074625080284478, 12.38517204345165152999796760101

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