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Properties

Label 2-13e2-13.10-c3-0-22
Degree $2$
Conductor $169$
Sign $0.425 + 0.905i$
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  + (−4.70 + 2.71i)2-s + (−0.837 − 1.45i)3-s + (10.7 − 18.5i)4-s − 7.70i·5-s + (7.87 + 4.54i)6-s + (13.0 + 7.51i)7-s + 73.1i·8-s + (12.0 − 20.9i)9-s + (20.9 + 36.2i)10-s + (−2.17 + 1.25i)11-s − 35.9·12-s − 81.5·14-s + (−11.1 + 6.45i)15-s + (−112. − 194. i)16-s + (−1.03 + 1.78i)17-s + 131. i·18-s + ⋯
L(s)  = 1  + (−1.66 + 0.959i)2-s + (−0.161 − 0.279i)3-s + (1.34 − 2.32i)4-s − 0.689i·5-s + (0.535 + 0.309i)6-s + (0.702 + 0.405i)7-s + 3.23i·8-s + (0.448 − 0.776i)9-s + (0.661 + 1.14i)10-s + (−0.0595 + 0.0344i)11-s − 0.865·12-s − 1.55·14-s + (−0.192 + 0.111i)15-s + (−1.75 − 3.04i)16-s + (−0.0147 + 0.0255i)17-s + 1.71i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.425 + 0.905i$
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: \((2,\ 169,\ (\ :3/2),\ 0.425 + 0.905i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.523434 - 0.332342i\)
\(L(\frac12)\) \(\approx\) \(0.523434 - 0.332342i\)
\(L(\frac{5}{2})\) 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 \( 1 + (4.70 - 2.71i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (0.837 + 1.45i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + 7.70iT - 125T^{2} \)
7 \( 1 + (-13.0 - 7.51i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (2.17 - 1.25i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (1.03 - 1.78i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (81.7 + 47.2i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-17.9 - 31.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-70.0 - 121. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 264. iT - 2.97e4T^{2} \)
37 \( 1 + (-222. + 128. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (341. - 197. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-128. + 222. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 415. iT - 1.03e5T^{2} \)
53 \( 1 + 504.T + 1.48e5T^{2} \)
59 \( 1 + (104. + 60.1i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-376. + 651. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (183. - 105. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (355. + 205. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 17.4iT - 3.89e5T^{2} \)
79 \( 1 + 174.T + 4.93e5T^{2} \)
83 \( 1 + 963. iT - 5.71e5T^{2} \)
89 \( 1 + (-413. + 238. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-906. - 523. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−11.82436594562261781434987241110, −10.89383783777247980563055712590, −9.717981290644055834157478304951, −8.896094071808658018368728374009, −8.175455318850599384074442886066, −7.07464783149111531679459901338, −6.12435765508398198708926820427, −4.90752528256255085121272767127, −1.82780013580335907180212356681, −0.53545262938832480013533459280, 1.45910556804147920731593790195, 2.80288403306023078775686518274, 4.37889257465983353209384421326, 6.68349768622499448410330598672, 7.72716519600444948363066755868, 8.463975308387774882863221743439, 9.765425525022240611214771227221, 10.68758369906479798707101456340, 10.86978304846552418842755344834, 12.02541882915715365455141522615

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