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Properties

Label 2-13e2-13.9-c3-0-12
Degree $2$
Conductor $169$
Sign $0.664 - 0.746i$
Analytic cond. $9.97132$
Root an. cond. $3.15774$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.5i)2-s + (−1 + 1.73i)3-s + (2.5 + 4.33i)4-s − 1.73·5-s + (1.73 + 3i)6-s + (−6.92 − 12i)7-s + 22.5·8-s + (11.5 + 19.9i)9-s + (−1.49 + 2.59i)10-s + (−6.92 + 12i)11-s − 10·12-s − 24·14-s + (1.73 − 2.99i)15-s + (−0.500 + 0.866i)16-s + (58.5 + 101. i)17-s + 39.8·18-s + ⋯
L(s)  = 1  + (0.306 − 0.530i)2-s + (−0.192 + 0.333i)3-s + (0.312 + 0.541i)4-s − 0.154·5-s + (0.117 + 0.204i)6-s + (−0.374 − 0.647i)7-s + 0.995·8-s + (0.425 + 0.737i)9-s + (−0.0474 + 0.0821i)10-s + (−0.189 + 0.328i)11-s − 0.240·12-s − 0.458·14-s + (0.0298 − 0.0516i)15-s + (−0.00781 + 0.0135i)16-s + (0.834 + 1.44i)17-s + 0.521·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.664 - 0.746i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.664 - 0.746i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.77473 + 0.796160i\)
\(L(\frac12)\) \(\approx\) \(1.77473 + 0.796160i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.866 + 1.5i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (1 - 1.73i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 1.73T + 125T^{2} \)
7 \( 1 + (6.92 + 12i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (6.92 - 12i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-58.5 - 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-57.1 - 99i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (39 - 67.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-70.5 + 122. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 155.T + 2.97e4T^{2} \)
37 \( 1 + (-71.8 + 124.5i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-135. + 235.5i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-52 - 90.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 301.T + 1.03e5T^{2} \)
53 \( 1 - 93T + 1.48e5T^{2} \)
59 \( 1 + (142. + 246i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (72.5 + 125. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (393. - 681i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (528. + 915i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 458.T + 3.89e5T^{2} \)
79 \( 1 - 1.27e3T + 4.93e5T^{2} \)
83 \( 1 + 789.T + 5.71e5T^{2} \)
89 \( 1 + (-488. + 846i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (100. + 174i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.39623942928793974657061602211, −11.54883796333243871922109179655, −10.36478935328565959691397164380, −10.00856824614306502642142863055, −8.004059712867094393146556608061, −7.50495917969218783065890384682, −5.91070358036158874848223804640, −4.33967047057970719492919851472, −3.52566074895650894001091441065, −1.77478074190564237779904645005, 0.890123206138796746847547626870, 2.86185262921755595038003357780, 4.75840455120789094673337462944, 5.87004057268646170482864223630, 6.75006598656477399595559965963, 7.68816290506712583437886042950, 9.241675016152584317428472020578, 10.07201731967082591347445622789, 11.45042478850239602538847388399, 12.10529919163493778372700463087

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