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Properties

Label 2-13e2-13.2-c2-0-17
Degree $2$
Conductor $169$
Sign $0.940 + 0.338i$
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  + (1.92 + 0.515i)2-s + (0.299 + 0.519i)3-s + (−0.0309 − 0.0178i)4-s + (5.65 − 5.65i)5-s + (0.308 + 1.15i)6-s + (1.96 − 0.526i)7-s + (−5.68 − 5.68i)8-s + (4.32 − 7.48i)9-s + (13.7 − 7.95i)10-s + (−3.76 + 14.0i)11-s − 0.0214i·12-s + 4.05·14-s + (4.62 + 1.24i)15-s + (−7.92 − 13.7i)16-s + (10.2 + 5.90i)17-s + (12.1 − 12.1i)18-s + ⋯
L(s)  = 1  + (0.961 + 0.257i)2-s + (0.0999 + 0.173i)3-s + (−0.00773 − 0.00446i)4-s + (1.13 − 1.13i)5-s + (0.0514 + 0.192i)6-s + (0.280 − 0.0752i)7-s + (−0.710 − 0.710i)8-s + (0.480 − 0.831i)9-s + (1.37 − 0.795i)10-s + (−0.342 + 1.27i)11-s − 0.00178i·12-s + 0.289·14-s + (0.308 + 0.0826i)15-s + (−0.495 − 0.858i)16-s + (0.601 + 0.347i)17-s + (0.675 − 0.675i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.940 + 0.338i$
Analytic conductor: \(4.60491\)
Root analytic conductor: \(2.14590\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :1),\ 0.940 + 0.338i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.57907 - 0.450333i\)
\(L(\frac12)\) \(\approx\) \(2.57907 - 0.450333i\)
\(L(2)\) 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 \( 1 + (-1.92 - 0.515i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (-0.299 - 0.519i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-5.65 + 5.65i)T - 25iT^{2} \)
7 \( 1 + (-1.96 + 0.526i)T + (42.4 - 24.5i)T^{2} \)
11 \( 1 + (3.76 - 14.0i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (-10.2 - 5.90i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.28 - 23.4i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (0.229 - 0.132i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (3.60 + 6.25i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (30.2 - 30.2i)T - 961iT^{2} \)
37 \( 1 + (3.53 - 13.1i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + (33.6 + 9.00i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (35.0 + 20.2i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (9.87 + 9.87i)T + 2.20e3iT^{2} \)
53 \( 1 - 77.1T + 2.80e3T^{2} \)
59 \( 1 + (-34.8 + 9.32i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-15.4 + 26.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-60.4 - 16.1i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + (2.23 + 8.34i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (-23.3 - 23.3i)T + 5.32e3iT^{2} \)
79 \( 1 + 49.8T + 6.24e3T^{2} \)
83 \( 1 + (60.7 - 60.7i)T - 6.88e3iT^{2} \)
89 \( 1 + (22.9 - 85.6i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (-8.75 - 32.6i)T + (-8.14e3 + 4.70e3i)T^{2} \)
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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

−12.56830287034276709006325616048, −12.21609342870950138266742342185, −10.00774344372366703959544937752, −9.722247914219591918664422937848, −8.541554252623482829451525207021, −6.90808866962831181460650825295, −5.64957993331227634182105829143, −4.94767628470369084357770893369, −3.79599397496935433433917736045, −1.53555852341215507486665569055, 2.30516251637738829845594621606, 3.30604739676516240317823356619, 5.06250210441236490017404549784, 5.87576735630183454618574877818, 7.16811059324068860971981135498, 8.492086121034477012279525946456, 9.772594621487086659906018557689, 10.88270450775610632024779267822, 11.53686150661793959961383003711, 13.15039869545051386122149859649

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