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Properties

Label 1-265-265.8-r0-0-0
Degree $1$
Conductor $265$
Sign $-0.747 + 0.664i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 265 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(265\)    =    \(5 \cdot 53\)
Sign: $-0.747 + 0.664i$
Analytic conductor: \(1.23065\)
Root analytic conductor: \(1.23065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{265} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 265,\ (0:\ ),\ -0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04203457709 + 0.1105673975i\)
\(L(\frac12)\) \(\approx\) \(0.04203457709 + 0.1105673975i\)
\(L(1)\) \(\approx\) \(0.6940972106 - 0.1179441298i\)
\(L(1)\) \(\approx\) \(0.6940972106 - 0.1179441298i\)

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

−25.16900040145611562445847314805, −24.67742749397402761285757985284, −23.73078759131626162241322181806, −23.012610507940998121751905355318, −22.11030155085971574105408282929, −21.144027707103593630645731945953, −19.663444581491512139168491206200, −18.82070190560343411469047807913, −18.155200185181322704829822081880, −16.98186037863782848915631580078, −16.22280360490996775848766586468, −15.23637888514278551545018309523, −13.960144911375437147443545048085, −13.54791669871812617034506093947, −12.42742252739223634595872144355, −11.7843615529691440049320360724, −9.9153075503844473347343611135, −8.75124224572136372861055035524, −7.97900857248644244102309628201, −6.76115839507615284508100827666, −6.23790965913175126258337920891, −5.06801117554493974892251986090, −3.488758154313304076979092236543, −2.38866071545127671187338344327, −0.064983002314095102191706522120, 2.269697267322677966521441169771, 3.26020721919097473117902779267, 4.30074643044212842255766514779, 5.173044356996930891992899749535, 6.45503849714352031871842792468, 8.21556426538180529209337426401, 9.36762805822763292402617308970, 10.25853799478077231108638741781, 10.612661854251206684981369503292, 12.08269436398328892655136655976, 12.92588231196768353648293906114, 13.88099311925463249423739729814, 15.10309276718990851587595139083, 15.58668705615094033718547009734, 17.03680323655207226790162755461, 17.8466056379500347425246750739, 19.32902307140592891187355961143, 19.91526429472203117071267710021, 20.6140859602616390041252219079, 21.70445070232230057043888461338, 22.31822474032873532039621716009, 23.05841621062297530969813116425, 24.0231649000564967828525700536, 25.64471122976941092274154657293, 26.24922553253278147436574183722

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