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Properties

Label 2-13e2-13.4-c3-0-31
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  + (0.379 + 0.219i)2-s + (1.84 − 3.19i)3-s + (−3.90 − 6.76i)4-s − 17.8i·5-s + (1.39 − 0.807i)6-s + (−4.70 + 2.71i)7-s − 6.93i·8-s + (6.71 + 11.6i)9-s + (3.90 − 6.76i)10-s + (−19.4 − 11.2i)11-s − 28.7·12-s − 2.38·14-s + (−56.8 − 32.8i)15-s + (−29.7 + 51.4i)16-s + (33.9 + 58.8i)17-s + 5.88i·18-s + ⋯
L(s)  = 1  + (0.134 + 0.0775i)2-s + (0.354 − 0.614i)3-s + (−0.487 − 0.845i)4-s − 1.59i·5-s + (0.0951 − 0.0549i)6-s + (−0.254 + 0.146i)7-s − 0.306i·8-s + (0.248 + 0.430i)9-s + (0.123 − 0.213i)10-s + (−0.532 − 0.307i)11-s − 0.692·12-s − 0.0455·14-s + (−0.978 − 0.564i)15-s + (−0.464 + 0.804i)16-s + (0.484 + 0.839i)17-s + 0.0770i·18-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

\(L(2)\) \(\approx\) \(0.174164 - 1.35650i\)
\(L(\frac12)\) \(\approx\) \(0.174164 - 1.35650i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.379 - 0.219i)T + (4 + 6.92i)T^{2} \)
3 \( 1 + (-1.84 + 3.19i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 17.8iT - 125T^{2} \)
7 \( 1 + (4.70 - 2.71i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (19.4 + 11.2i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-33.9 - 58.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (69.9 - 40.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-70.2 + 121. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-53.3 + 92.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 276. iT - 2.97e4T^{2} \)
37 \( 1 + (3.71 + 2.14i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (197. + 113. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-13.7 - 23.8i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 318. iT - 1.03e5T^{2} \)
53 \( 1 + 67.6T + 1.48e5T^{2} \)
59 \( 1 + (-252. + 145. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (331. + 574. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-368. - 212. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (132. - 76.4i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 117. iT - 3.89e5T^{2} \)
79 \( 1 - 202.T + 4.93e5T^{2} \)
83 \( 1 - 336. iT - 5.71e5T^{2} \)
89 \( 1 + (-621. - 359. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-657. + 379. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.34791054502377512944446581732, −10.70533986301151897689044271032, −9.730374371702346737826897823216, −8.611178312104393647016433005944, −8.047819401770306457504084953177, −6.31194459289698257801390581183, −5.25197695492614323637041861974, −4.27398906573708969725014337863, −1.91661180956621047180821609984, −0.58806893503262310754640350266, 2.89103166605888257262451750419, 3.48058862792448028688463929424, 4.86600695110030073202686501069, 6.73733722794321228252968714957, 7.45271905388234097428342862177, 8.829748338625735941415778542549, 9.837886482748586357173637494504, 10.66410812646118217766861878871, 11.74894020361106731204049050773, 12.84034952153685261040379284816

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