Properties

Label 1-605-605.339-r0-0-0
Degree $1$
Conductor $605$
Sign $-0.790 + 0.612i$
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  + (0.466 − 0.884i)2-s + (−0.309 + 0.951i)3-s + (−0.564 − 0.825i)4-s + (0.696 + 0.717i)6-s + (0.254 − 0.967i)7-s + (−0.993 + 0.113i)8-s + (−0.809 − 0.587i)9-s + (0.959 − 0.281i)12-s + (−0.941 − 0.336i)13-s + (−0.736 − 0.676i)14-s + (−0.362 + 0.931i)16-s + (−0.974 − 0.226i)17-s + (−0.897 + 0.441i)18-s + (0.516 + 0.856i)19-s + (0.841 + 0.540i)21-s + ⋯
L(s)  = 1  + (0.466 − 0.884i)2-s + (−0.309 + 0.951i)3-s + (−0.564 − 0.825i)4-s + (0.696 + 0.717i)6-s + (0.254 − 0.967i)7-s + (−0.993 + 0.113i)8-s + (−0.809 − 0.587i)9-s + (0.959 − 0.281i)12-s + (−0.941 − 0.336i)13-s + (−0.736 − 0.676i)14-s + (−0.362 + 0.931i)16-s + (−0.974 − 0.226i)17-s + (−0.897 + 0.441i)18-s + (0.516 + 0.856i)19-s + (0.841 + 0.540i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.790 + 0.612i$
Analytic conductor: \(2.80960\)
Root analytic conductor: \(2.80960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 605,\ (0:\ ),\ -0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05178348162 - 0.1513112676i\)
\(L(\frac12)\) \(\approx\) \(-0.05178348162 - 0.1513112676i\)
\(L(1)\) \(\approx\) \(0.7329993168 - 0.2872031614i\)
\(L(1)\) \(\approx\) \(0.7329993168 - 0.2872031614i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.466 + 0.884i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (-0.254 + 0.967i)T \)
13 \( 1 + (0.941 + 0.336i)T \)
17 \( 1 + (0.974 + 0.226i)T \)
19 \( 1 + (-0.516 - 0.856i)T \)
23 \( 1 + (0.841 - 0.540i)T \)
29 \( 1 + (0.921 + 0.389i)T \)
31 \( 1 + (-0.610 - 0.791i)T \)
37 \( 1 + (-0.0285 - 0.999i)T \)
41 \( 1 + (0.985 + 0.170i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (0.897 + 0.441i)T \)
53 \( 1 + (-0.362 - 0.931i)T \)
59 \( 1 + (0.985 - 0.170i)T \)
61 \( 1 + (0.466 + 0.884i)T \)
67 \( 1 + (-0.142 - 0.989i)T \)
71 \( 1 + (-0.0855 + 0.996i)T \)
73 \( 1 + (0.774 + 0.633i)T \)
79 \( 1 + (0.870 - 0.491i)T \)
83 \( 1 + (-0.998 + 0.0570i)T \)
89 \( 1 + (0.654 + 0.755i)T \)
97 \( 1 + (0.198 - 0.980i)T \)
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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

−23.88448454367605983416747841261, −22.74240891429211939823656352057, −22.15350476954788572177498786778, −21.54700894730641152885429805849, −20.206765303392613959077475625908, −19.24303704411974563549468607244, −18.26302729987524579980598423418, −17.80699902169273966964926992066, −16.9523255816016205882672110682, −16.06869809722261885386656150098, −15.08662396405498926448470568932, −14.409166901660675446129436419234, −13.43416463721342565154687293198, −12.73111329709592251582190075114, −11.91954878949686043281457670369, −11.273409638985359095384803048477, −9.5384261127040513739797706118, −8.64716367107000078003037505450, −7.8409175024954597473718238791, −6.940810556539849782292613975617, −6.185649830588390336551520600554, −5.29799995754650152006561903698, −4.48684311843270950401741262917, −2.872163855488119596088460193733, −2.00682927891413294545904218631, 0.06735909342285364513548780969, 1.652972903500127874558739093040, 3.03618277192045255569978675444, 3.89756260025189081929297581296, 4.70276964162494473428632394095, 5.44731701873356468166064474568, 6.617000708143964075482790488, 7.99129749905778241354032358626, 9.20559042442083733387044985637, 10.0856240364059725268266097935, 10.48134354764508860115870649989, 11.54781744846423336921150043430, 12.0878428062543704749285650397, 13.38724249342701680821653714561, 14.064787449753686066089001998370, 14.933225466608314982836027523719, 15.69660457800354906969776686841, 16.87020513087725091430990998662, 17.5278324752193191873700721613, 18.46312823204168326899715168941, 19.78703348224490133175111497309, 20.185436309241517059338911573052, 20.877735367879001470806028534867, 21.84397072431084810411049456483, 22.37581622261254305783890583030

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