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Properties

Label 2-13e2-13.10-c3-0-20
Degree $2$
Conductor $169$
Sign $0.967 + 0.252i$
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  + (3.46 − 2i)2-s + (−1 − 1.73i)3-s + (3.99 − 6.92i)4-s + 17i·5-s + (−6.92 − 3.99i)6-s + (17.3 + 10i)7-s + (11.5 − 19.9i)9-s + (34 + 58.8i)10-s + (27.7 − 16i)11-s − 15.9·12-s + 80·14-s + (29.4 − 17i)15-s + (31.9 + 55.4i)16-s + (−6.5 + 11.2i)17-s − 92i·18-s + (−25.9 − 15i)19-s + ⋯
L(s)  = 1  + (1.22 − 0.707i)2-s + (−0.192 − 0.333i)3-s + (0.499 − 0.866i)4-s + 1.52i·5-s + (−0.471 − 0.272i)6-s + (0.935 + 0.539i)7-s + (0.425 − 0.737i)9-s + (1.07 + 1.86i)10-s + (0.759 − 0.438i)11-s − 0.384·12-s + 1.52·14-s + (0.506 − 0.292i)15-s + (0.499 + 0.866i)16-s + (−0.0927 + 0.160i)17-s − 1.20i·18-s + (−0.313 − 0.181i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.967 + 0.252i$
Analytic conductor: \(9.97132\)
Root analytic conductor: \(3.15774\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ 0.967 + 0.252i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.36275 - 0.431752i\)
\(L(\frac12)\) \(\approx\) \(3.36275 - 0.431752i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.46 + 2i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (1 + 1.73i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 - 17iT - 125T^{2} \)
7 \( 1 + (-17.3 - 10i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-27.7 + 16i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (6.5 - 11.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (25.9 + 15i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-39 - 67.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (98.5 + 170. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 74iT - 2.97e4T^{2} \)
37 \( 1 + (-196. + 113.5i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (142. - 82.5i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (78 - 135. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 162iT - 1.03e5T^{2} \)
53 \( 1 - 93T + 1.48e5T^{2} \)
59 \( 1 + (748. + 432i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (72.5 - 125. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-746. + 431i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (566. + 327i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 215iT - 3.89e5T^{2} \)
79 \( 1 + 76T + 4.93e5T^{2} \)
83 \( 1 - 628iT - 5.71e5T^{2} \)
89 \( 1 + (-230. + 133i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (206. + 119i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.13687757413320869896724752614, −11.38686262471289680451561470937, −10.96737475469685343588857812177, −9.511688941655934429260573859276, −7.933881390920548303463260050649, −6.60916279389742401079830992922, −5.79331407752267733878280277254, −4.23142765820583406878319203939, −3.14978761307355954195363194354, −1.85042139776134295975923511880, 1.37234244543934425469486742792, 4.11360195887226238013533721709, 4.71314616758000778788612895226, 5.40454543706290499328685267004, 6.93545204156817504678880127090, 8.016155787730277047372686460105, 9.177115583880786930432269119586, 10.46021395257273857043030722160, 11.75064434942650027101840188461, 12.66533173773120425053973979774

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