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Properties

Label 1-260-260.7-r1-0-0
Degree $1$
Conductor $260$
Sign $0.0557 - 0.998i$
Analytic cond. $27.9408$
Root an. cond. $27.9408$
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  + (0.866 − 0.5i)3-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − i·31-s + (0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.866 − 0.5i)41-s + (0.866 + 0.5i)43-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s + (0.866 − 0.5i)11-s + (0.866 + 0.5i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (−0.866 + 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − i·31-s + (0.5 − 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.866 − 0.5i)41-s + (0.866 + 0.5i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0557 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0557 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(260\)    =    \(2^{2} \cdot 5 \cdot 13\)
Sign: $0.0557 - 0.998i$
Analytic conductor: \(27.9408\)
Root analytic conductor: \(27.9408\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{260} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 260,\ (1:\ ),\ 0.0557 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.082104629 - 1.969115855i\)
\(L(\frac12)\) \(\approx\) \(2.082104629 - 1.969115855i\)
\(L(1)\) \(\approx\) \(1.489612934 - 0.5928177049i\)
\(L(1)\) \(\approx\) \(1.489612934 - 0.5928177049i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.866 - 0.5i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 - T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−25.6888165900975786166420230788, −25.12011073536699745688011991649, −24.3859201765798405849477434010, −23.01917832168750042161800621502, −22.040909057816395834514178876772, −21.2219261080870796770258535115, −20.508937689041147240214038765296, −19.451167818614300717044295689919, −18.70502914723781652937307914367, −17.5793800641801867498970742458, −16.41860063250843570475769480350, −15.47707507582957527131861744679, −14.57793898768292433121263961614, −14.06035159168539424278638175408, −12.58851817695515701670273623729, −11.77149336583541019828886085822, −10.40973544167008436044498981509, −9.49942807461727782656789491916, −8.57836567182475766041940261705, −7.75704344684916374978258363903, −6.31909051787402016574625971542, −4.97075667958247345723760919903, −3.99146479632547844758146398677, −2.69495797952331663077130901780, −1.620829025734895824789997321, 0.84649959551242823592445366488, 1.945944693560030199286967668419, 3.46099137119749701294810228447, 4.276456201668533361753918077178, 6.00854940772366059449467329106, 7.11088036803295227623608167762, 8.0028484320768963812240374991, 8.91170966368642737967016626773, 10.03872232815100834126117685171, 11.190713688104321188599274150048, 12.33062578430841475029840590081, 13.34400269550641538031773964569, 14.25552010622772509002544400720, 14.75815783342555821634010002793, 16.14904249724124829900723892958, 17.22553925313270079782667177460, 18.036791970359894651265039311198, 19.355088536170507747736171539515, 19.66792852571526214571944127928, 20.83836683668052808727683235323, 21.51421978894782471066454841388, 22.87002273991379015596136975158, 23.95525527190320710567924544599, 24.338696595755289127782796854452, 25.609928525496249969760490508028

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