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Properties

Label 1-3328-3328.1867-r0-0-0
Degree $1$
Conductor $3328$
Sign $0.576 - 0.817i$
Analytic cond. $15.4551$
Root an. cond. $15.4551$
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  + (−0.0980 − 0.995i)3-s + (0.471 − 0.881i)5-s + (0.980 + 0.195i)7-s + (−0.980 + 0.195i)9-s + (−0.773 + 0.634i)11-s + (−0.923 − 0.382i)15-s + (−0.382 − 0.923i)17-s + (0.956 + 0.290i)19-s + (0.0980 − 0.995i)21-s + (0.831 + 0.555i)23-s + (−0.555 − 0.831i)25-s + (0.290 + 0.956i)27-s + (0.773 + 0.634i)29-s + (−0.707 + 0.707i)31-s + (0.707 + 0.707i)33-s + ⋯
L(s)  = 1  + (−0.0980 − 0.995i)3-s + (0.471 − 0.881i)5-s + (0.980 + 0.195i)7-s + (−0.980 + 0.195i)9-s + (−0.773 + 0.634i)11-s + (−0.923 − 0.382i)15-s + (−0.382 − 0.923i)17-s + (0.956 + 0.290i)19-s + (0.0980 − 0.995i)21-s + (0.831 + 0.555i)23-s + (−0.555 − 0.831i)25-s + (0.290 + 0.956i)27-s + (0.773 + 0.634i)29-s + (−0.707 + 0.707i)31-s + (0.707 + 0.707i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.576 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.576 - 0.817i$
Analytic conductor: \(15.4551\)
Root analytic conductor: \(15.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3328,\ (0:\ ),\ 0.576 - 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.764102375 - 0.9146930458i\)
\(L(\frac12)\) \(\approx\) \(1.764102375 - 0.9146930458i\)
\(L(1)\) \(\approx\) \(1.148220867 - 0.4353260717i\)
\(L(1)\) \(\approx\) \(1.148220867 - 0.4353260717i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.0980 - 0.995i)T \)
5 \( 1 + (0.471 - 0.881i)T \)
7 \( 1 + (0.980 + 0.195i)T \)
11 \( 1 + (-0.773 + 0.634i)T \)
17 \( 1 + (-0.382 - 0.923i)T \)
19 \( 1 + (0.956 + 0.290i)T \)
23 \( 1 + (0.831 + 0.555i)T \)
29 \( 1 + (0.773 + 0.634i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (0.290 + 0.956i)T \)
41 \( 1 + (0.831 + 0.555i)T \)
43 \( 1 + (-0.0980 + 0.995i)T \)
47 \( 1 + (-0.382 - 0.923i)T \)
53 \( 1 + (0.773 - 0.634i)T \)
59 \( 1 + (0.881 + 0.471i)T \)
61 \( 1 + (0.995 - 0.0980i)T \)
67 \( 1 + (0.0980 + 0.995i)T \)
71 \( 1 + (0.195 - 0.980i)T \)
73 \( 1 + (0.980 - 0.195i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (-0.290 + 0.956i)T \)
89 \( 1 + (-0.555 - 0.831i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−18.86799082593711358210165122481, −18.14290050879942326023448296960, −17.51114591432239117535458166170, −17.02010138998498664790038175561, −16.07758472318209110558400186471, −15.44330087101910858235961532885, −14.81550486014459370573660280585, −14.2122604278756596212548734274, −13.66608410335053950554580776761, −12.74905382433824942772284691527, −11.56799869144011289350498444807, −11.03213907934331207133174294407, −10.666083950826768833635082107982, −9.95212493939890566881727830066, −9.11947919547230901078537995106, −8.37139074573980172641217934177, −7.63544687137012143421570660413, −6.737972937260979498535921851673, −5.6960706503010674370710093540, −5.40761668024031942363539467688, −4.37680276446642905379393268829, −3.6909864487317166824122712306, −2.74664105855467551046637434191, −2.17029886707003643034905581880, −0.76110832005261832707379895512, 0.888003525735496832868908169336, 1.49263534824686809184049102406, 2.30541566079655414442417704865, 3.05985679001460662725423306078, 4.52834708179340593643489539258, 5.25590141090074094825567409439, 5.45280751447042143818275059422, 6.712298026504982231238237698082, 7.35202628387625162432014343771, 8.07944133625924422237110686702, 8.66215189411281500827057906090, 9.44225088268958805708247267209, 10.27694529561777873986357147987, 11.37977613591554951630335380354, 11.71023884005729767949539660068, 12.61539361816598243949019050414, 13.07974308888092083907832303728, 13.7978377610517154910408383494, 14.38253560890221442645422270969, 15.23032798271950165495671138304, 16.16908451730667335596596100235, 16.72673259410224117381185603213, 17.76992089374747127817462792568, 17.91502785021755965056596908813, 18.40911966006139699124198396453

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