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Properties

Label 1-1205-1205.1088-r0-0-0
Degree $1$
Conductor $1205$
Sign $-0.127 + 0.991i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (0.707 + 0.707i)2-s + (0.891 − 0.453i)3-s + i·4-s + (0.951 + 0.309i)6-s + (0.972 − 0.233i)7-s + (−0.707 + 0.707i)8-s + (0.587 − 0.809i)9-s + (−0.382 + 0.923i)11-s + (0.453 + 0.891i)12-s + (0.0784 + 0.996i)13-s + (0.852 + 0.522i)14-s − 16-s + (−0.522 + 0.852i)17-s + (0.987 − 0.156i)18-s + (−0.923 − 0.382i)19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (0.891 − 0.453i)3-s + i·4-s + (0.951 + 0.309i)6-s + (0.972 − 0.233i)7-s + (−0.707 + 0.707i)8-s + (0.587 − 0.809i)9-s + (−0.382 + 0.923i)11-s + (0.453 + 0.891i)12-s + (0.0784 + 0.996i)13-s + (0.852 + 0.522i)14-s − 16-s + (−0.522 + 0.852i)17-s + (0.987 − 0.156i)18-s + (−0.923 − 0.382i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.127 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.127 + 0.991i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1088, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ -0.127 + 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.067376683 + 2.350500777i\)
\(L(\frac12)\) \(\approx\) \(2.067376683 + 2.350500777i\)
\(L(1)\) \(\approx\) \(1.821441132 + 0.9405385543i\)
\(L(1)\) \(\approx\) \(1.821441132 + 0.9405385543i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (0.891 - 0.453i)T \)
7 \( 1 + (0.972 - 0.233i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (0.0784 + 0.996i)T \)
17 \( 1 + (-0.522 + 0.852i)T \)
19 \( 1 + (-0.923 - 0.382i)T \)
23 \( 1 + (-0.649 + 0.760i)T \)
29 \( 1 + (0.156 + 0.987i)T \)
31 \( 1 + (0.852 + 0.522i)T \)
37 \( 1 + (-0.0784 + 0.996i)T \)
41 \( 1 + (-0.156 + 0.987i)T \)
43 \( 1 + (0.233 - 0.972i)T \)
47 \( 1 + (0.987 - 0.156i)T \)
53 \( 1 + (-0.156 - 0.987i)T \)
59 \( 1 + (-0.891 - 0.453i)T \)
61 \( 1 + (0.891 - 0.453i)T \)
67 \( 1 + (0.987 + 0.156i)T \)
71 \( 1 + (0.233 - 0.972i)T \)
73 \( 1 + (0.0784 - 0.996i)T \)
79 \( 1 + (0.891 + 0.453i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−20.950603150857945117489103463259, −20.44780895698574074760419800781, −19.67170423795529175654386466792, −18.80212157059315460011401259704, −18.295061444967756661339194271249, −17.16528934586286664679011155788, −15.78088265852439782828855850889, −15.532537889314489714136901112529, −14.52403133782488851627238752081, −14.03721835972149061880830872829, −13.3158326830810362449425017327, −12.49964616618533939052781661535, −11.48209356276078610939751188380, −10.72437873038616580893961655416, −10.20416595102976027806598545583, −9.0958973121467383599801360847, −8.3781520053618139182230870583, −7.62768015199896549674138089703, −6.167816655643171121948523302909, −5.35594745970481354074183506045, −4.472398641460760108450900354446, −3.81930902050927154387893512623, −2.58411167439981945220194878363, −2.3256249184069237961084451791, −0.853212750381755416722270823636, 1.70569960431255122041143507351, 2.26368834293282653654282301603, 3.54866892640865945595461709683, 4.36404821012708101690685084120, 4.97118640825510333905494695381, 6.396972905285019590395805042731, 6.927627881385207301951965216879, 7.82154861871723246141618055237, 8.412203419805697984675917974604, 9.15247493215969190644179861095, 10.362124117755246784953662504533, 11.49713501255997241181382115819, 12.28759102787915166169101169352, 13.019429126086614240576059389909, 13.820540458359699741176725306466, 14.33730625413853137801327514863, 15.183007988803395736769621221221, 15.53015526169187860183781565667, 16.79604571723771966598497428301, 17.58079171181005110461588880781, 18.07358884515268863844662081582, 19.11941199520328576924063409216, 20.06405698601475728113304697511, 20.685242632514108288600571862855, 21.44785371970980990573079679009

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