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Properties

Label 1-805-805.172-r0-0-0
Degree $1$
Conductor $805$
Sign $0.991 + 0.126i$
Analytic cond. $3.73840$
Root an. cond. $3.73840$
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  + (−0.814 + 0.580i)2-s + (0.971 − 0.235i)3-s + (0.327 − 0.945i)4-s + (−0.654 + 0.755i)6-s + (0.281 + 0.959i)8-s + (0.888 − 0.458i)9-s + (−0.580 + 0.814i)11-s + (0.0950 − 0.995i)12-s + (−0.989 + 0.142i)13-s + (−0.786 − 0.618i)16-s + (0.189 − 0.981i)17-s + (−0.458 + 0.888i)18-s + (0.981 − 0.189i)19-s − i·22-s + (0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.814 + 0.580i)2-s + (0.971 − 0.235i)3-s + (0.327 − 0.945i)4-s + (−0.654 + 0.755i)6-s + (0.281 + 0.959i)8-s + (0.888 − 0.458i)9-s + (−0.580 + 0.814i)11-s + (0.0950 − 0.995i)12-s + (−0.989 + 0.142i)13-s + (−0.786 − 0.618i)16-s + (0.189 − 0.981i)17-s + (−0.458 + 0.888i)18-s + (0.981 − 0.189i)19-s − i·22-s + (0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(3.73840\)
Root analytic conductor: \(3.73840\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 805,\ (0:\ ),\ 0.991 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.371611041 + 0.08745126355i\)
\(L(\frac12)\) \(\approx\) \(1.371611041 + 0.08745126355i\)
\(L(1)\) \(\approx\) \(1.025678052 + 0.1112639609i\)
\(L(1)\) \(\approx\) \(1.025678052 + 0.1112639609i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.814 + 0.580i)T \)
3 \( 1 + (0.971 - 0.235i)T \)
11 \( 1 + (-0.580 + 0.814i)T \)
13 \( 1 + (-0.989 + 0.142i)T \)
17 \( 1 + (0.189 - 0.981i)T \)
19 \( 1 + (0.981 - 0.189i)T \)
29 \( 1 + (0.654 - 0.755i)T \)
31 \( 1 + (0.723 + 0.690i)T \)
37 \( 1 + (0.458 + 0.888i)T \)
41 \( 1 + (0.841 - 0.540i)T \)
43 \( 1 + (0.281 - 0.959i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (0.371 + 0.928i)T \)
59 \( 1 + (0.786 - 0.618i)T \)
61 \( 1 + (-0.235 + 0.971i)T \)
67 \( 1 + (0.0950 + 0.995i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (0.945 + 0.327i)T \)
79 \( 1 + (0.928 + 0.371i)T \)
83 \( 1 + (-0.540 + 0.841i)T \)
89 \( 1 + (0.723 - 0.690i)T \)
97 \( 1 + (-0.540 - 0.841i)T \)
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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

−21.70749972893050761156219754293, −21.38147251969867509731735691698, −20.522347179676988143484727685029, −19.63488727732293047374130173247, −19.314742608881568553022033762369, −18.3763178566213548366579384407, −17.63094418654855954222047617634, −16.49342644222084618401227185789, −15.99569337290711501409510750602, −14.964898372854717406382753083189, −14.10176930618922637243872809738, −13.10664584079688472666577620737, −12.49004076592059101012173927633, −11.35833962676421727418347651639, −10.465932283561907808641212713929, −9.78270882987754343136102775168, −9.05161606147452948452285536368, −7.95334875870433728604258278611, −7.78689245812102267690425449075, −6.46664445421302295472838949933, −5.01724890613684075360603742530, −3.85322640912744054928705740539, −3.01106502621649294245189931366, −2.28726768927469573663941532737, −1.04898063828485709245374027802, 0.92011228116653002689279075325, 2.21013910523753646880397216880, 2.8597139199856836493033287004, 4.50193301209599481775176301110, 5.3040234600429858045172450226, 6.725869537307340147360428230625, 7.34958349667400469644808352836, 7.95701379655590202282001126280, 8.9468688593995284416536708031, 9.79890197159723089375782819408, 10.14330706228713394152100794308, 11.61012408031912756620010194920, 12.443127248645727340372555968673, 13.699288940474900385061581201283, 14.18601761841617010613203638937, 15.20486968376457668913360030267, 15.63416362199855059835999825563, 16.59403588438809958907590419127, 17.702328088567987505178012091511, 18.16702751405121097132939794127, 19.07126967106814598022153548283, 19.73515512992322394865402432116, 20.440735463183517495195045024236, 21.09497285961948582509296969460, 22.40586893232311750199745116407

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