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Properties

Label 2-13e2-13.12-c7-0-69
Degree $2$
Conductor $169$
Sign $-0.832 + 0.554i$
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  − 6.08i·2-s + 25.6·3-s + 90.9·4-s + 442. i·5-s − 155. i·6-s − 761. i·7-s − 1.33e3i·8-s − 1.53e3·9-s + 2.69e3·10-s − 6.11e3i·11-s + 2.33e3·12-s − 4.63e3·14-s + 1.13e4i·15-s + 3.54e3·16-s − 3.75e4·17-s + 9.30e3i·18-s + ⋯
L(s)  = 1  − 0.537i·2-s + 0.548·3-s + 0.710·4-s + 1.58i·5-s − 0.294i·6-s − 0.839i·7-s − 0.919i·8-s − 0.699·9-s + 0.850·10-s − 1.38i·11-s + 0.389·12-s − 0.451·14-s + 0.867i·15-s + 0.216·16-s − 1.85·17-s + 0.376i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.832 + 0.554i$
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: \((2,\ 169,\ (\ :7/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.507731204\)
\(L(\frac12)\) \(\approx\) \(1.507731204\)
\(L(\frac{9}{2})\) 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 \( 1 + 6.08iT - 128T^{2} \)
3 \( 1 - 25.6T + 2.18e3T^{2} \)
5 \( 1 - 442. iT - 7.81e4T^{2} \)
7 \( 1 + 761. iT - 8.23e5T^{2} \)
11 \( 1 + 6.11e3iT - 1.94e7T^{2} \)
17 \( 1 + 3.75e4T + 4.10e8T^{2} \)
19 \( 1 - 3.39e3iT - 8.93e8T^{2} \)
23 \( 1 - 2.98e4T + 3.40e9T^{2} \)
29 \( 1 + 4.22e4T + 1.72e10T^{2} \)
31 \( 1 + 1.24e5iT - 2.75e10T^{2} \)
37 \( 1 + 1.42e5iT - 9.49e10T^{2} \)
41 \( 1 + 7.14e4iT - 1.94e11T^{2} \)
43 \( 1 - 1.27e4T + 2.71e11T^{2} \)
47 \( 1 + 4.37e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.01e6T + 1.17e12T^{2} \)
59 \( 1 + 1.75e6iT - 2.48e12T^{2} \)
61 \( 1 + 1.69e6T + 3.14e12T^{2} \)
67 \( 1 - 3.41e6iT - 6.06e12T^{2} \)
71 \( 1 + 7.94e5iT - 9.09e12T^{2} \)
73 \( 1 + 3.45e6iT - 1.10e13T^{2} \)
79 \( 1 - 6.86e6T + 1.92e13T^{2} \)
83 \( 1 + 8.04e6iT - 2.71e13T^{2} \)
89 \( 1 - 1.04e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.12e6iT - 8.07e13T^{2} \)
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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

−11.01883215499156510589234734589, −10.53470947382804893407822784745, −9.151227873606653693468743795214, −7.83017300376142524617741639311, −6.83837184544700636474125966248, −6.08764458354798407934828102684, −3.78521356703745734755941469579, −3.02392806469875763319021746088, −2.15963351939169272562687605739, −0.31046053490682974188110448157, 1.68332094634570567120522687748, 2.58491151263156800527334546718, 4.56286045338036693677559561532, 5.43402158103082561362038619090, 6.66781976316938354468471022592, 7.933958511188083648795183485527, 8.787637668857675938594935898361, 9.344931979027043292945031046061, 11.05466799468092322355906249018, 12.06889390572161552111644043656

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