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Properties

Label 1-605-605.169-r0-0-0
Degree $1$
Conductor $605$
Sign $-0.977 + 0.211i$
Analytic cond. $2.80960$
Root an. cond. $2.80960$
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  + (−0.516 + 0.856i)2-s + (0.809 + 0.587i)3-s + (−0.466 − 0.884i)4-s + (−0.921 + 0.389i)6-s + (−0.610 + 0.791i)7-s + (0.998 + 0.0570i)8-s + (0.309 + 0.951i)9-s + (0.142 − 0.989i)12-s + (0.985 − 0.170i)13-s + (−0.362 − 0.931i)14-s + (−0.564 + 0.825i)16-s + (−0.993 + 0.113i)17-s + (−0.974 − 0.226i)18-s + (−0.870 + 0.491i)19-s + (−0.959 + 0.281i)21-s + ⋯
L(s)  = 1  + (−0.516 + 0.856i)2-s + (0.809 + 0.587i)3-s + (−0.466 − 0.884i)4-s + (−0.921 + 0.389i)6-s + (−0.610 + 0.791i)7-s + (0.998 + 0.0570i)8-s + (0.309 + 0.951i)9-s + (0.142 − 0.989i)12-s + (0.985 − 0.170i)13-s + (−0.362 − 0.931i)14-s + (−0.564 + 0.825i)16-s + (−0.993 + 0.113i)17-s + (−0.974 − 0.226i)18-s + (−0.870 + 0.491i)19-s + (−0.959 + 0.281i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.977 + 0.211i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ -0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1145012048 + 1.071561023i\)
\(L(\frac12)\) \(\approx\) \(0.1145012048 + 1.071561023i\)
\(L(1)\) \(\approx\) \(0.6710833811 + 0.6564380468i\)
\(L(1)\) \(\approx\) \(0.6710833811 + 0.6564380468i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.516 + 0.856i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
7 \( 1 + (-0.610 + 0.791i)T \)
13 \( 1 + (0.985 - 0.170i)T \)
17 \( 1 + (-0.993 + 0.113i)T \)
19 \( 1 + (-0.870 + 0.491i)T \)
23 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (0.198 + 0.980i)T \)
31 \( 1 + (0.897 - 0.441i)T \)
37 \( 1 + (-0.696 - 0.717i)T \)
41 \( 1 + (0.0855 + 0.996i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (-0.974 + 0.226i)T \)
53 \( 1 + (0.564 + 0.825i)T \)
59 \( 1 + (0.0855 - 0.996i)T \)
61 \( 1 + (0.516 + 0.856i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (-0.736 - 0.676i)T \)
73 \( 1 + (-0.941 + 0.336i)T \)
79 \( 1 + (-0.254 + 0.967i)T \)
83 \( 1 + (0.0285 - 0.999i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.774 - 0.633i)T \)
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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

−22.7649355256929967801468987392, −21.535434831298625458339340159018, −20.74478200379953289366874412207, −20.185951702100219537395133565194, −19.23435355456869041147209888249, −18.97835387265950402365787570902, −17.81106452862014464097140172196, −17.21830212951000742562804999606, −16.09662602916149406737463989071, −15.12746225026308175369327553158, −13.67517747412806366620525538344, −13.47074407483570744383580930469, −12.65806227063093914983470473989, −11.58923462985746489205616647580, −10.66236432965336097638456870918, −9.816323898775135473519383310355, −8.78585249456901458716121329706, −8.35309953248685260053650892878, −7.05857421569377028207982353610, −6.54632573363599208732816299087, −4.52337932928704830449248878673, −3.67495949903258220258668516845, −2.803035842047337906243746228268, −1.75916320996433913129134640346, −0.59528389802380064771218439982, 1.615150464364381172000243738603, 2.86375137322508912358895541454, 4.036890108119335733827095957263, 5.06728944783471712605105487781, 6.11185737954923622438890955223, 6.94743879260578736324624296268, 8.28078366587915170878847289038, 8.69310010627333617225374254720, 9.47506569565814484640566944269, 10.37206219240288759887136137707, 11.2183309976714887748806779134, 12.91339311715765872512720052972, 13.45296691222943046927426197458, 14.60388840552566747766796012084, 15.20023722453691769854573812486, 15.89984981853601140003131267400, 16.496105454010795431073263838681, 17.63141553160285635904185194560, 18.57511110272611414530159112315, 19.2268078674452333467293003430, 19.923948631757187655516783362674, 20.98580462915056597203149116938, 21.83299436110049335126901114493, 22.7286129673092861426865685785, 23.4842846309361084412297800014

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