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Properties

Label 1-13e2-169.103-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: $\chi_{169} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ 0.986 - 0.166i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.014945780 - 0.1689482568i\)
\(L(\frac12)\) \(\approx\) \(2.014945780 - 0.1689482568i\)
\(L(1)\) \(\approx\) \(1.794320132 - 0.07728412045i\)
\(L(1)\) \(\approx\) \(1.794320132 - 0.07728412045i\)

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

−27.594591869746365166344425207051, −26.39869605654501052347231000175, −25.26796447367229336136712721800, −24.62267634835613561931563781674, −23.79695236192931519250019947091, −22.82440121067668943962026486023, −22.13249863374212142339854133646, −20.92606972344818937281997967069, −19.74796848114621022901913974988, −18.93271814661741556564970207023, −17.83119464943600547782430116818, −16.75611538821891800859648413180, −15.16780347231495996444649892890, −14.7237608768939581028648076936, −13.7266195488079980697302131266, −12.62074955457923762403053302667, −11.48393923792593830598108387996, −11.221116893700298865293974023944, −8.80770298899375000432065452744, −7.719296400007236658792849391676, −6.78903447342518776160295467122, −5.92026309972537241677486855401, −4.383196532886697306645197600774, −2.99020539342198741672308470672, −1.94584268849531270470723952991, 1.61420192845068322317751915267, 3.52302135447067085607865401671, 4.38973014567394388846425063931, 4.996525889476614890842779538850, 6.57937572893229940730400795767, 8.10564229925296881509237629802, 9.26765053869391343745583987742, 10.71670035019898336217561404305, 11.34369299577596876629226006556, 12.495485456688208052961626260385, 13.67049897584849147414942043248, 14.80058533354995006040111602612, 15.33032500564560820745052994360, 16.65243757784475793863761274566, 17.16359401449584919870774412540, 19.43578790916003635699237492454, 20.05044150304511013841313176563, 20.8545147812969948060140224099, 21.61170542365980367739505111906, 22.71927622581085744299886859374, 23.55223284082028691428732391346, 24.45737749231430178635647545197, 25.44729994542951131269138903107, 26.821615752842184445768202220365, 27.71219210137291961914163800658

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