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Properties

Label 1-1305-1305.799-r0-0-0
Degree $1$
Conductor $1305$
Sign $-0.956 - 0.290i$
Analytic cond. $6.06039$
Root an. cond. $6.06039$
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  + (−0.365 + 0.930i)2-s + (−0.733 − 0.680i)4-s + (0.733 − 0.680i)7-s + (0.900 − 0.433i)8-s + (0.0747 + 0.997i)11-s + (−0.826 + 0.563i)13-s + (0.365 + 0.930i)14-s + (0.0747 + 0.997i)16-s − 17-s + (−0.222 + 0.974i)19-s + (−0.955 − 0.294i)22-s + (−0.365 − 0.930i)23-s + (−0.222 − 0.974i)26-s − 28-s + (−0.988 + 0.149i)31-s + (−0.955 − 0.294i)32-s + ⋯
L(s)  = 1  + (−0.365 + 0.930i)2-s + (−0.733 − 0.680i)4-s + (0.733 − 0.680i)7-s + (0.900 − 0.433i)8-s + (0.0747 + 0.997i)11-s + (−0.826 + 0.563i)13-s + (0.365 + 0.930i)14-s + (0.0747 + 0.997i)16-s − 17-s + (−0.222 + 0.974i)19-s + (−0.955 − 0.294i)22-s + (−0.365 − 0.930i)23-s + (−0.222 − 0.974i)26-s − 28-s + (−0.988 + 0.149i)31-s + (−0.955 − 0.294i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $-0.956 - 0.290i$
Analytic conductor: \(6.06039\)
Root analytic conductor: \(6.06039\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1305} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1305,\ (0:\ ),\ -0.956 - 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05845145764 + 0.3933868282i\)
\(L(\frac12)\) \(\approx\) \(-0.05845145764 + 0.3933868282i\)
\(L(1)\) \(\approx\) \(0.6200679599 + 0.3211315545i\)
\(L(1)\) \(\approx\) \(0.6200679599 + 0.3211315545i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.365 + 0.930i)T \)
7 \( 1 + (0.733 - 0.680i)T \)
11 \( 1 + (0.0747 + 0.997i)T \)
13 \( 1 + (-0.826 + 0.563i)T \)
17 \( 1 - T \)
19 \( 1 + (-0.222 + 0.974i)T \)
23 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + (-0.988 + 0.149i)T \)
37 \( 1 + (0.900 - 0.433i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.988 + 0.149i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (-0.0747 + 0.997i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.623 + 0.781i)T \)
79 \( 1 + (0.826 + 0.563i)T \)
83 \( 1 + (-0.955 + 0.294i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (-0.955 + 0.294i)T \)
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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

−20.442350978363156531049988629113, −19.81526215653706579325850953241, −19.094419927634099400987842180046, −18.364524940842814083517642209865, −17.56436117349842974282711494630, −17.17868067501858943645574307832, −15.98751316742877728574718233844, −15.20182918345833430971529379824, −14.24037174910558394153986520448, −13.48152824356120975128762412609, −12.695239956670478536773056864218, −11.88411325340198617672032774890, −11.12593340012340911328423516647, −10.72301346345611438979499438072, −9.40844621573561940473305287128, −9.0082187240014508793084631950, −8.082001982821283065142600369390, −7.4177520918367448035077908385, −6.02721229294308308585061220019, −5.0938336258412184251715305802, −4.3481412325920069777753621847, −3.1740308605979452179539707089, −2.44661656250023155243260297221, −1.535398578548331651643458779204, −0.1757390032294728918161614475, 1.40887123115814379901457125052, 2.25113467764921136247152076169, 4.160906948288230402218391199532, 4.435749811696446517824895523121, 5.423434902820153065665727735909, 6.52790221819557743337978164967, 7.18379095885902986594649828362, 7.8513189017151610431980730596, 8.70715074211510811728192457631, 9.60673964267007818146769782756, 10.29252959468520827232250078043, 11.09736728531632524382321682279, 12.209379185911362661564723108071, 13.07259770231630085089196695409, 13.996631339301293748577057944742, 14.726751391153413356689849189067, 15.02046317721077781527620589662, 16.34115042683060307845310920354, 16.69302377498133112125861468784, 17.66575954446682207598893664497, 18.00455223457933797219759535382, 18.95332907374928908180041731446, 19.91645932385266882334662512150, 20.35321133262064350954742923781, 21.48987815232712672385984978894

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