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Properties

Label 1-605-605.409-r1-0-0
Degree $1$
Conductor $605$
Sign $0.999 + 0.0337i$
Analytic cond. $65.0162$
Root an. cond. $65.0162$
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  + (0.610 + 0.791i)2-s + (−0.309 + 0.951i)3-s + (−0.254 + 0.967i)4-s + (−0.941 + 0.336i)6-s + (0.993 + 0.113i)7-s + (−0.921 + 0.389i)8-s + (−0.809 − 0.587i)9-s + (−0.841 − 0.540i)12-s + (−0.362 − 0.931i)13-s + (0.516 + 0.856i)14-s + (−0.870 − 0.491i)16-s + (0.696 + 0.717i)17-s + (−0.0285 − 0.999i)18-s + (−0.897 − 0.441i)19-s + (−0.415 + 0.909i)21-s + ⋯
L(s)  = 1  + (0.610 + 0.791i)2-s + (−0.309 + 0.951i)3-s + (−0.254 + 0.967i)4-s + (−0.941 + 0.336i)6-s + (0.993 + 0.113i)7-s + (−0.921 + 0.389i)8-s + (−0.809 − 0.587i)9-s + (−0.841 − 0.540i)12-s + (−0.362 − 0.931i)13-s + (0.516 + 0.856i)14-s + (−0.870 − 0.491i)16-s + (0.696 + 0.717i)17-s + (−0.0285 − 0.999i)18-s + (−0.897 − 0.441i)19-s + (−0.415 + 0.909i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.999 + 0.0337i$
Analytic conductor: \(65.0162\)
Root analytic conductor: \(65.0162\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (1:\ ),\ 0.999 + 0.0337i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.321666233 + 0.02230698559i\)
\(L(\frac12)\) \(\approx\) \(1.321666233 + 0.02230698559i\)
\(L(1)\) \(\approx\) \(0.9430721407 + 0.6680989143i\)
\(L(1)\) \(\approx\) \(0.9430721407 + 0.6680989143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.610 + 0.791i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (0.993 + 0.113i)T \)
13 \( 1 + (-0.362 - 0.931i)T \)
17 \( 1 + (0.696 + 0.717i)T \)
19 \( 1 + (-0.897 - 0.441i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
29 \( 1 + (0.985 - 0.170i)T \)
31 \( 1 + (-0.998 - 0.0570i)T \)
37 \( 1 + (-0.774 - 0.633i)T \)
41 \( 1 + (0.564 - 0.825i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (0.0285 - 0.999i)T \)
53 \( 1 + (0.870 - 0.491i)T \)
59 \( 1 + (-0.564 - 0.825i)T \)
61 \( 1 + (-0.610 + 0.791i)T \)
67 \( 1 + (0.959 + 0.281i)T \)
71 \( 1 + (-0.466 - 0.884i)T \)
73 \( 1 + (-0.736 + 0.676i)T \)
79 \( 1 + (-0.974 + 0.226i)T \)
83 \( 1 + (0.198 + 0.980i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (-0.0855 - 0.996i)T \)
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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

−23.22303929659309034878795468336, −21.87375303455882294274236338086, −21.34306363362654784969036898623, −20.37773837663804351656149528341, −19.56236528927443166622322641436, −18.798948992608361873651811426679, −18.121584651305046167533986688662, −17.29059052433496730750189481436, −16.26623807580252917141043994175, −14.85387810646135618511389810174, −14.19620773504658095099382376377, −13.613659555276923939310776133086, −12.532370570963793683305174363573, −11.85621406175250906914048864602, −11.28284792021208037803338011596, −10.34450307266680404780971800376, −9.17895440247555113829258307671, −8.08949215406800625871325559495, −7.09782976976091425364167386777, −6.07576453414234075106945418109, −5.14910462092216775262666672236, −4.32316915897933501012333302023, −2.9412288835113126913194068534, −1.8518001690002355145389253260, −1.215511231054148787915354465730, 0.269657294434542133705726577655, 2.35560027558020670263876879432, 3.59011942779587133379559761717, 4.430130387706953755630097370460, 5.27372077192658071775548357297, 5.90315521778825990619866155788, 7.12229620785371779625730606726, 8.2929711077291264305192223921, 8.73984446246100913778734214080, 10.16748364807941458425086106520, 10.901253249512985632410183503408, 12.009467918033560447987065474256, 12.644797732822381706324592565632, 13.96571978592886267009786363680, 14.742196082096583981939225339320, 15.175302303552408152308316245446, 16.07300115816683191634035491206, 17.03214532943041900137875461779, 17.47414100757685732623366574535, 18.357355066671524202371686825326, 19.87614295544223899722653434085, 20.77588594893254016177111496122, 21.43440812553545684401609425534, 22.015168935161049290070551802428, 22.92699661090500211687500664685

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