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Properties

Label 1-265-265.167-r0-0-0
Degree $1$
Conductor $265$
Sign $0.928 + 0.371i$
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  + (−0.354 + 0.935i)2-s + (−0.120 − 0.992i)3-s + (−0.748 − 0.663i)4-s + (0.970 + 0.239i)6-s + (0.935 + 0.354i)7-s + (0.885 − 0.464i)8-s + (−0.970 + 0.239i)9-s + (−0.568 + 0.822i)11-s + (−0.568 + 0.822i)12-s + (0.663 + 0.748i)13-s + (−0.663 + 0.748i)14-s + (0.120 + 0.992i)16-s + (0.464 − 0.885i)17-s + (0.120 − 0.992i)18-s + (−0.663 − 0.748i)19-s + ⋯
L(s)  = 1  + (−0.354 + 0.935i)2-s + (−0.120 − 0.992i)3-s + (−0.748 − 0.663i)4-s + (0.970 + 0.239i)6-s + (0.935 + 0.354i)7-s + (0.885 − 0.464i)8-s + (−0.970 + 0.239i)9-s + (−0.568 + 0.822i)11-s + (−0.568 + 0.822i)12-s + (0.663 + 0.748i)13-s + (−0.663 + 0.748i)14-s + (0.120 + 0.992i)16-s + (0.464 − 0.885i)17-s + (0.120 − 0.992i)18-s + (−0.663 − 0.748i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(265\)    =    \(5 \cdot 53\)
Sign: $0.928 + 0.371i$
Analytic conductor: \(1.23065\)
Root analytic conductor: \(1.23065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{265} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 265,\ (0:\ ),\ 0.928 + 0.371i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9662504446 + 0.1861071631i\)
\(L(\frac12)\) \(\approx\) \(0.9662504446 + 0.1861071631i\)
\(L(1)\) \(\approx\) \(0.8800193615 + 0.1419182491i\)
\(L(1)\) \(\approx\) \(0.8800193615 + 0.1419182491i\)

Euler product

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

−26.14727318441430054678301047361, −25.14007951855187589370050468453, −23.507796056534145471618600445051, −22.963788601257038110289466127939, −21.668785300582645118040601229764, −21.11375205983285908185998150754, −20.59268337719334567842563841035, −19.46604483598098935052198054902, −18.46484282118031994706068523230, −17.380137859938555622109563704891, −16.806978065515556345418537260974, −15.581318878452006151987430831047, −14.48111632676330372650832037798, −13.53111123462138590922498132981, −12.3469183385207455473838582449, −11.05227618976537519895006653832, −10.77437541069524801661817492509, −9.81429786522556021214882439519, −8.38839982802151523755140464596, −8.120206933726932248009075407463, −5.892134476042458049096018341451, −4.78336175139450671384038223500, −3.801462949402731791580100801483, −2.78970796684707509562907575110, −1.07703605259690024317230286707, 1.12098213807061551953099279964, 2.39428989397704541567421105902, 4.59160695823789408138017552627, 5.44109997478450942114991098733, 6.665940392730947369489415186876, 7.41032217757102857551963740488, 8.37077964893251027798663913335, 9.19005510376974962778097124968, 10.70514579780412454562931945571, 11.72068028152120917760900670312, 12.93176918196667143151762964646, 13.83203008804565769653126434778, 14.68487901971575650093004438075, 15.622202443173765086142468141071, 16.84076263816456381766860608806, 17.63915403648689261717369388461, 18.3819173265937181122017604156, 18.9624564608043521533351536822, 20.202743563681968944409766779525, 21.33485127604635572050484799814, 22.715659580054311166536694889152, 23.51540488193924810901988914958, 24.00756843157673612465764197807, 25.09042994625264624993490827324, 25.54149674801085754951438633350

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