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Properties

Label 1-13e2-169.105-r0-0-0
Degree $1$
Conductor $169$
Sign $-0.986 - 0.166i$
Analytic cond. $0.784832$
Root an. cond. $0.784832$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.970 − 0.239i)2-s + (0.120 − 0.992i)3-s + (0.885 + 0.464i)4-s + (0.568 − 0.822i)5-s + (−0.354 + 0.935i)6-s + (−0.748 − 0.663i)7-s + (−0.748 − 0.663i)8-s + (−0.970 − 0.239i)9-s + (−0.748 + 0.663i)10-s + (−0.970 + 0.239i)11-s + (0.568 − 0.822i)12-s + (0.568 + 0.822i)14-s + (−0.748 − 0.663i)15-s + (0.568 + 0.822i)16-s + (−0.748 − 0.663i)17-s + (0.885 + 0.464i)18-s + ⋯
L(s)  = 1  + (−0.970 − 0.239i)2-s + (0.120 − 0.992i)3-s + (0.885 + 0.464i)4-s + (0.568 − 0.822i)5-s + (−0.354 + 0.935i)6-s + (−0.748 − 0.663i)7-s + (−0.748 − 0.663i)8-s + (−0.970 − 0.239i)9-s + (−0.748 + 0.663i)10-s + (−0.970 + 0.239i)11-s + (0.568 − 0.822i)12-s + (0.568 + 0.822i)14-s + (−0.748 − 0.663i)15-s + (0.568 + 0.822i)16-s + (−0.748 − 0.663i)17-s + (0.885 + 0.464i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.986 - 0.166i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ -0.986 - 0.166i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04684593783 - 0.5587037508i\)
\(L(\frac12)\) \(\approx\) \(0.04684593783 - 0.5587037508i\)
\(L(1)\) \(\approx\) \(0.4702348160 - 0.4270755752i\)
\(L(1)\) \(\approx\) \(0.4702348160 - 0.4270755752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (-0.970 - 0.239i)T \)
3 \( 1 + (0.120 - 0.992i)T \)
5 \( 1 + (0.568 - 0.822i)T \)
7 \( 1 + (-0.748 - 0.663i)T \)
11 \( 1 + (-0.970 + 0.239i)T \)
17 \( 1 + (-0.748 - 0.663i)T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + (-0.970 - 0.239i)T \)
31 \( 1 + (-0.354 + 0.935i)T \)
37 \( 1 + (-0.354 + 0.935i)T \)
41 \( 1 + (0.120 - 0.992i)T \)
43 \( 1 + (-0.354 - 0.935i)T \)
47 \( 1 + (0.885 - 0.464i)T \)
53 \( 1 + (-0.748 - 0.663i)T \)
59 \( 1 + (0.568 - 0.822i)T \)
61 \( 1 + (-0.748 + 0.663i)T \)
67 \( 1 + (0.885 - 0.464i)T \)
71 \( 1 + (0.120 - 0.992i)T \)
73 \( 1 + (-0.970 + 0.239i)T \)
79 \( 1 + (0.885 - 0.464i)T \)
83 \( 1 + (0.120 + 0.992i)T \)
89 \( 1 + T \)
97 \( 1 + (0.568 + 0.822i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.064831859686738777672311711212, −26.69528861256239627619664795827, −26.32944174162826013340466900737, −25.57103669131064739253753074685, −24.600155009466140085740486900446, −23.07958709884468864107060847599, −22.05198913230482357796891779303, −21.247599330928753591492376899718, −20.19366846685784293807367325439, −19.06149575248825848228952861621, −18.28264049966145027969603095033, −17.21616344165260690503979146276, −16.12906899433927437376025177517, −15.385100331120555527668846407467, −14.606806788161078621222603722264, −13.14644359289650280898793749713, −11.3427789151020566877961680325, −10.62244025967653013410354913356, −9.64519269260633574298597106665, −8.97974611586246229658617917448, −7.58619488591752458821879481497, −6.19529016767813713632753856513, −5.39786486951356377877627403091, −3.2310370062479135462161270296, −2.36537918907374508224377269129, 0.58132552610903325239765974375, 1.93805835945170957887258274566, 3.169346791446382646398921753743, 5.35071787479851751212790773928, 6.75421823972400754009483827970, 7.52638027604034043047626865747, 8.75209945467928473478047677319, 9.60852069832059301455936691684, 10.80144980223519908247453958942, 12.11958108422110711028090110508, 13.02338356790365178018126372172, 13.70502625107856088364088230130, 15.59295542156338673939578179420, 16.61026595290830721937027171266, 17.44775377427845026446345512801, 18.27999948433194175918558108689, 19.22729334767563726451817800445, 20.3539521052020306210333396182, 20.61550623785930531763889561048, 22.27435093467006023378011877009, 23.59793809816640411096894653968, 24.47972269879286153652170319527, 25.33275030212830020826599769174, 26.04813607101260456766038604343, 27.02671245302098964995191895914

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