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Properties

Label 2-13e2-13.12-c7-0-44
Degree $2$
Conductor $169$
Sign $-0.277 - 0.960i$
Analytic cond. $52.7930$
Root an. cond. $7.26588$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.679i·2-s + 62.4·3-s + 127.·4-s + 439. i·5-s + 42.4i·6-s + 1.30e3i·7-s + 173. i·8-s + 1.71e3·9-s − 298.·10-s + 574. i·11-s + 7.96e3·12-s − 884.·14-s + 2.74e4i·15-s + 1.62e4·16-s + 1.12e4·17-s + 1.16e3i·18-s + ⋯
L(s)  = 1  + 0.0600i·2-s + 1.33·3-s + 0.996·4-s + 1.57i·5-s + 0.0801i·6-s + 1.43i·7-s + 0.119i·8-s + 0.783·9-s − 0.0943·10-s + 0.130i·11-s + 1.33·12-s − 0.0861·14-s + 2.09i·15-s + 0.989·16-s + 0.556·17-s + 0.0470i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(52.7930\)
Root analytic conductor: \(7.26588\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 169,\ (\ :7/2),\ -0.277 - 0.960i)\)

Particular Values

\(L(4)\) \(\approx\) \(4.364931168\)
\(L(\frac12)\) \(\approx\) \(4.364931168\)
\(L(\frac{9}{2})\) 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 \( 1 - 0.679iT - 128T^{2} \)
3 \( 1 - 62.4T + 2.18e3T^{2} \)
5 \( 1 - 439. iT - 7.81e4T^{2} \)
7 \( 1 - 1.30e3iT - 8.23e5T^{2} \)
11 \( 1 - 574. iT - 1.94e7T^{2} \)
17 \( 1 - 1.12e4T + 4.10e8T^{2} \)
19 \( 1 + 4.10e4iT - 8.93e8T^{2} \)
23 \( 1 - 4.42e4T + 3.40e9T^{2} \)
29 \( 1 + 1.45e5T + 1.72e10T^{2} \)
31 \( 1 - 1.22e5iT - 2.75e10T^{2} \)
37 \( 1 - 4.20e4iT - 9.49e10T^{2} \)
41 \( 1 + 8.76e4iT - 1.94e11T^{2} \)
43 \( 1 - 7.49e5T + 2.71e11T^{2} \)
47 \( 1 + 9.40e5iT - 5.06e11T^{2} \)
53 \( 1 + 9.24e5T + 1.17e12T^{2} \)
59 \( 1 + 6.32e5iT - 2.48e12T^{2} \)
61 \( 1 + 6.44e4T + 3.14e12T^{2} \)
67 \( 1 + 1.94e6iT - 6.06e12T^{2} \)
71 \( 1 - 4.72e6iT - 9.09e12T^{2} \)
73 \( 1 - 1.92e6iT - 1.10e13T^{2} \)
79 \( 1 + 1.50e6T + 1.92e13T^{2} \)
83 \( 1 + 1.87e6iT - 2.71e13T^{2} \)
89 \( 1 - 4.37e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.77e6iT - 8.07e13T^{2} \)
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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

−11.58281570568572309631898587626, −10.86769415478333527641450731479, −9.646384107169439617651997374753, −8.683821605932493888748796583076, −7.52677736023996231087455014014, −6.79434945879940001105142766874, −5.61185193615562784602982553979, −3.33056681055872930494605730692, −2.71764911251562136727618335815, −2.04823502054747035473090639917, 0.906863534468228296462009163150, 1.77569013931811317115581596312, 3.32725826919622788498347917240, 4.25673496345806576617069962156, 5.83358179415957682640647214564, 7.53354971571117789724717197621, 7.894460781282075737452949829100, 9.095060904004543685195907615462, 9.993984046130347293686461612051, 11.13537051044486418551071057085

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