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Properties

Label 2-13e2-13.4-c3-0-0
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: $\chi_{169} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :3/2),\ -0.967 + 0.252i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0194985 - 0.151866i\)
\(L(\frac12)\) \(\approx\) \(0.0194985 - 0.151866i\)
\(L(\frac{5}{2})\) 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

−12.57998725580713967687749619927, −11.23509152968745386557155805763, −10.73153460798287497712961859034, −10.05202325178981471494805147695, −9.147633935025717020481531939898, −7.84204876759055956765639938549, −6.83875942143023981378150222726, −5.42685636239814820828882099831, −3.23488663651536556725432799365, −2.32061267747781760103864705700, 0.11977577245846564232444661477, 1.21806263970241569955795356761, 3.96009558417502354075370650205, 5.56497464752801068374341290045, 6.80222687969741269464423094826, 7.65691460233254945002174343382, 8.726663677637035073899286257173, 9.571902769493888238445679887066, 10.19018200157696172684799014454, 11.93352266431654794102173104980

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