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Properties

Label 1-405-405.92-r0-0-0
Degree $1$
Conductor $405$
Sign $0.634 + 0.772i$
Analytic cond. $1.88081$
Root an. cond. $1.88081$
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  + (−0.998 + 0.0581i)2-s + (0.993 − 0.116i)4-s + (0.802 + 0.597i)7-s + (−0.984 + 0.173i)8-s + (0.686 + 0.727i)11-s + (−0.448 − 0.893i)13-s + (−0.835 − 0.549i)14-s + (0.973 − 0.230i)16-s + (0.342 + 0.939i)17-s + (0.939 + 0.342i)19-s + (−0.727 − 0.686i)22-s + (−0.802 + 0.597i)23-s + (0.5 + 0.866i)26-s + (0.866 + 0.5i)28-s + (−0.835 + 0.549i)29-s + ⋯
L(s)  = 1  + (−0.998 + 0.0581i)2-s + (0.993 − 0.116i)4-s + (0.802 + 0.597i)7-s + (−0.984 + 0.173i)8-s + (0.686 + 0.727i)11-s + (−0.448 − 0.893i)13-s + (−0.835 − 0.549i)14-s + (0.973 − 0.230i)16-s + (0.342 + 0.939i)17-s + (0.939 + 0.342i)19-s + (−0.727 − 0.686i)22-s + (−0.802 + 0.597i)23-s + (0.5 + 0.866i)26-s + (0.866 + 0.5i)28-s + (−0.835 + 0.549i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.772i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.634 + 0.772i$
Analytic conductor: \(1.88081\)
Root analytic conductor: \(1.88081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ 0.634 + 0.772i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8372173128 + 0.3958076912i\)
\(L(\frac12)\) \(\approx\) \(0.8372173128 + 0.3958076912i\)
\(L(1)\) \(\approx\) \(0.7838367980 + 0.1498182866i\)
\(L(1)\) \(\approx\) \(0.7838367980 + 0.1498182866i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.998 + 0.0581i)T \)
7 \( 1 + (0.802 + 0.597i)T \)
11 \( 1 + (0.686 + 0.727i)T \)
13 \( 1 + (-0.448 - 0.893i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
19 \( 1 + (0.939 + 0.342i)T \)
23 \( 1 + (-0.802 + 0.597i)T \)
29 \( 1 + (-0.835 + 0.549i)T \)
31 \( 1 + (0.396 - 0.918i)T \)
37 \( 1 + (-0.642 + 0.766i)T \)
41 \( 1 + (0.0581 - 0.998i)T \)
43 \( 1 + (0.957 + 0.286i)T \)
47 \( 1 + (0.918 - 0.396i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.686 + 0.727i)T \)
61 \( 1 + (-0.993 - 0.116i)T \)
67 \( 1 + (0.549 - 0.835i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + (0.984 - 0.173i)T \)
79 \( 1 + (0.0581 + 0.998i)T \)
83 \( 1 + (0.998 - 0.0581i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (0.230 + 0.973i)T \)
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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

−24.455700274474901329435334198006, −23.685660239812852735519916858664, −22.34634206580817968834731197759, −21.38171653477555221471634847759, −20.58160387851045573795767045350, −19.82127845923303269316666956633, −18.93408454134529484302978770701, −18.1110496280137600101405952490, −17.233673867187851424638077318506, −16.52830955794924138632508708875, −15.74425285159207539718524909551, −14.36276461007009493209166767898, −13.925664199072858933734766659058, −12.1959991714889654714817706783, −11.52609472186182840356790982402, −10.76485102079755301700038422909, −9.62475277174672144029542245797, −8.91449696710188067171224948018, −7.781361601512753288398997780105, −7.10862847025674470084612345114, −5.988151916323136958696075509226, −4.61616234707783250008662398866, −3.316352091304121567984583698727, −1.98597418176455651526818385364, −0.84774559556482899078503168608, 1.33655462820609306959613421823, 2.256152950909579291384808593983, 3.65121680902788904153548583461, 5.24462127148616718011904408122, 6.09796549665114585423274182995, 7.46642153847736605156236890945, 8.00390014814951361766572318526, 9.12022207671965488878326461036, 9.91864466404070670949906434915, 10.876790528352201667988988296670, 11.92944005927781022915572455557, 12.45239451566769751549226772612, 14.13790066982704392356356560307, 15.05469609534123016076974068823, 15.58147744329996742477048293017, 16.9159739642341572695893801107, 17.50316906272434018354594697789, 18.25011764901389366762818521586, 19.140313135909793648834350292342, 20.12031073926031829604944930360, 20.682052773858288088893716397982, 21.77719517144300058376156565421, 22.63283240785028039005317188332, 24.062036216150311353827683692099, 24.50609303645816312935310747211

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