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Properties

Label 1-265-265.162-r0-0-0
Degree $1$
Conductor $265$
Sign $-0.0671 - 0.997i$
Analytic cond. $1.23065$
Root an. cond. $1.23065$
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  + (−0.885 − 0.464i)2-s + (−0.354 + 0.935i)3-s + (0.568 + 0.822i)4-s + (0.748 − 0.663i)6-s + (0.464 − 0.885i)7-s + (−0.120 − 0.992i)8-s + (−0.748 − 0.663i)9-s + (0.970 − 0.239i)11-s + (−0.970 + 0.239i)12-s + (−0.822 − 0.568i)13-s + (−0.822 + 0.568i)14-s + (−0.354 + 0.935i)16-s + (−0.992 − 0.120i)17-s + (0.354 + 0.935i)18-s + (−0.822 − 0.568i)19-s + ⋯
L(s)  = 1  + (−0.885 − 0.464i)2-s + (−0.354 + 0.935i)3-s + (0.568 + 0.822i)4-s + (0.748 − 0.663i)6-s + (0.464 − 0.885i)7-s + (−0.120 − 0.992i)8-s + (−0.748 − 0.663i)9-s + (0.970 − 0.239i)11-s + (−0.970 + 0.239i)12-s + (−0.822 − 0.568i)13-s + (−0.822 + 0.568i)14-s + (−0.354 + 0.935i)16-s + (−0.992 − 0.120i)17-s + (0.354 + 0.935i)18-s + (−0.822 − 0.568i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0671 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0671 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(265\)    =    \(5 \cdot 53\)
Sign: $-0.0671 - 0.997i$
Analytic conductor: \(1.23065\)
Root analytic conductor: \(1.23065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{265} (162, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 265,\ (0:\ ),\ -0.0671 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3364392589 - 0.3598356476i\)
\(L(\frac12)\) \(\approx\) \(0.3364392589 - 0.3598356476i\)
\(L(1)\) \(\approx\) \(0.5657139731 - 0.1126143575i\)
\(L(1)\) \(\approx\) \(0.5657139731 - 0.1126143575i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
53 \( 1 \)
good2 \( 1 + (-0.885 - 0.464i)T \)
3 \( 1 + (-0.354 + 0.935i)T \)
7 \( 1 + (0.464 - 0.885i)T \)
11 \( 1 + (0.970 - 0.239i)T \)
13 \( 1 + (-0.822 - 0.568i)T \)
17 \( 1 + (-0.992 - 0.120i)T \)
19 \( 1 + (-0.822 - 0.568i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.970 - 0.239i)T \)
31 \( 1 + (0.239 - 0.970i)T \)
37 \( 1 + (0.935 + 0.354i)T \)
41 \( 1 + (-0.239 - 0.970i)T \)
43 \( 1 + (0.935 - 0.354i)T \)
47 \( 1 + (-0.663 - 0.748i)T \)
59 \( 1 + (-0.748 + 0.663i)T \)
61 \( 1 + (0.992 - 0.120i)T \)
67 \( 1 + (0.568 + 0.822i)T \)
71 \( 1 + (-0.935 + 0.354i)T \)
73 \( 1 + (0.120 - 0.992i)T \)
79 \( 1 + (-0.464 - 0.885i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.120 + 0.992i)T \)
97 \( 1 + (0.663 - 0.748i)T \)
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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

−25.87545575901026851281604624951, −24.9852438333950840150636900872, −24.47317301552400098132535923053, −23.74581458230110622743224434294, −22.56747545917646395025423000452, −21.59412850650090520662299549411, −20.07330603104940268874438999972, −19.407069378156283185571664568764, −18.56277612420048266317293029097, −17.74220033910539977755101684760, −17.095679153980510966594370757380, −16.10656286579724679875439885283, −14.77856594271001224836339112573, −14.261093320993136770483259562534, −12.6494860924749793868152340207, −11.76089267719334061037436556941, −11.0254286675394208248457432816, −9.558724978523564077326059374768, −8.65011690904151718548226500348, −7.74594903191986121094101890434, −6.64488967804064369471679171510, −5.98229695977330885414050997567, −4.690031738785869831618323948663, −2.313157131135898481570485454623, −1.60270621931144395971773712429, 0.46109231526655879068204575745, 2.23866232317777130029844052490, 3.75558864149957006203223607708, 4.49506068357767636708106662747, 6.174039115347527731595186769895, 7.32951849770283629156950473835, 8.52448293425747615872241504722, 9.478024498025793347055398401315, 10.33688995946857149963748600776, 11.16852647638364377139713169668, 11.86515117825855257854961534837, 13.23603319967131722908284933201, 14.61244267610857177439943872826, 15.52363054231784629056322440471, 16.705295705404366366508247782820, 17.20744280683166304060163233846, 17.90227236158886443283844886058, 19.41326102687119938683866774120, 20.12854428205883538745219468092, 20.77507107685902723266137891066, 21.98773019191450025658781103705, 22.35295455267009553204983038981, 23.8597133452112893905359121161, 24.79724530362674060124060987246, 26.09506321588674147943151597813

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