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Properties

Label 2-13e2-13.5-c4-0-36
Degree $2$
Conductor $169$
Sign $0.0129 + 0.999i$
Analytic cond. $17.4695$
Root an. cond. $4.17965$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.50 + 2.50i)2-s + 10.3·3-s + 3.46i·4-s + (−8.65 + 8.65i)5-s + (−25.8 + 25.8i)6-s + (−2.15 − 2.15i)7-s + (−48.7 − 48.7i)8-s + 25.7·9-s − 43.3i·10-s + (−146. − 146. i)11-s + 35.7i·12-s + 10.8·14-s + (−89.3 + 89.3i)15-s + 188.·16-s − 484. i·17-s + (−64.3 + 64.3i)18-s + ⋯
L(s)  = 1  + (−0.625 + 0.625i)2-s + 1.14·3-s + 0.216i·4-s + (−0.346 + 0.346i)5-s + (−0.718 + 0.718i)6-s + (−0.0440 − 0.0440i)7-s + (−0.761 − 0.761i)8-s + 0.317·9-s − 0.433i·10-s + (−1.21 − 1.21i)11-s + 0.248i·12-s + 0.0551·14-s + (−0.397 + 0.397i)15-s + 0.736·16-s − 1.67i·17-s + (−0.198 + 0.198i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0129 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0129 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $0.0129 + 0.999i$
Analytic conductor: \(17.4695\)
Root analytic conductor: \(4.17965\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (70, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :2),\ 0.0129 + 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4573238134\)
\(L(\frac12)\) \(\approx\) \(0.4573238134\)
\(L(3)\) 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 \( 1 + (2.50 - 2.50i)T - 16iT^{2} \)
3 \( 1 - 10.3T + 81T^{2} \)
5 \( 1 + (8.65 - 8.65i)T - 625iT^{2} \)
7 \( 1 + (2.15 + 2.15i)T + 2.40e3iT^{2} \)
11 \( 1 + (146. + 146. i)T + 1.46e4iT^{2} \)
17 \( 1 + 484. iT - 8.35e4T^{2} \)
19 \( 1 + (-13.1 + 13.1i)T - 1.30e5iT^{2} \)
23 \( 1 - 115. iT - 2.79e5T^{2} \)
29 \( 1 - 118.T + 7.07e5T^{2} \)
31 \( 1 + (1.05e3 - 1.05e3i)T - 9.23e5iT^{2} \)
37 \( 1 + (17.3 + 17.3i)T + 1.87e6iT^{2} \)
41 \( 1 + (-1.24e3 + 1.24e3i)T - 2.82e6iT^{2} \)
43 \( 1 + 2.07e3iT - 3.41e6T^{2} \)
47 \( 1 + (-507. - 507. i)T + 4.87e6iT^{2} \)
53 \( 1 + 2.09e3T + 7.89e6T^{2} \)
59 \( 1 + (-1.13e3 - 1.13e3i)T + 1.21e7iT^{2} \)
61 \( 1 + 3.28e3T + 1.38e7T^{2} \)
67 \( 1 + (865. - 865. i)T - 2.01e7iT^{2} \)
71 \( 1 + (720. - 720. i)T - 2.54e7iT^{2} \)
73 \( 1 + (5.16e3 + 5.16e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 8.14e3T + 3.89e7T^{2} \)
83 \( 1 + (-1.32e3 + 1.32e3i)T - 4.74e7iT^{2} \)
89 \( 1 + (-3.10e3 - 3.10e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (-4.68e3 + 4.68e3i)T - 8.85e7iT^{2} \)
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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

−11.81123615175563420282706480815, −10.69031979516754582403845739888, −9.308640524494386924491068112233, −8.694956768049048362160973496946, −7.73049356206277256035851086039, −7.15212335463666980979621918574, −5.50380964346883696323454769957, −3.47783806759487140029824252091, −2.79526703971144864540146254051, −0.17086544505895526218240294820, 1.77213128742932670471679281495, 2.76807708148264439928347702392, 4.36760529764976457912512079756, 5.89334661956309037063504600963, 7.73701757376883038580321716217, 8.348920787914409273442266006679, 9.370479315017959103645622842419, 10.16192765162024733747703921530, 11.09013607995446866829293527166, 12.43071420707387918096601821277

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