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Properties

Label 1-3328-3328.2283-r0-0-0
Degree $1$
Conductor $3328$
Sign $0.170 + 0.985i$
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  + (0.773 + 0.634i)3-s + (0.956 − 0.290i)5-s + (−0.195 + 0.980i)7-s + (0.195 + 0.980i)9-s + (0.995 + 0.0980i)11-s + (0.923 + 0.382i)15-s + (0.382 + 0.923i)17-s + (0.881 − 0.471i)19-s + (−0.773 + 0.634i)21-s + (0.555 − 0.831i)23-s + (0.831 − 0.555i)25-s + (−0.471 + 0.881i)27-s + (−0.995 + 0.0980i)29-s + (−0.707 + 0.707i)31-s + (0.707 + 0.707i)33-s + ⋯
L(s)  = 1  + (0.773 + 0.634i)3-s + (0.956 − 0.290i)5-s + (−0.195 + 0.980i)7-s + (0.195 + 0.980i)9-s + (0.995 + 0.0980i)11-s + (0.923 + 0.382i)15-s + (0.382 + 0.923i)17-s + (0.881 − 0.471i)19-s + (−0.773 + 0.634i)21-s + (0.555 − 0.831i)23-s + (0.831 − 0.555i)25-s + (−0.471 + 0.881i)27-s + (−0.995 + 0.0980i)29-s + (−0.707 + 0.707i)31-s + (0.707 + 0.707i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.170 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.170 + 0.985i$
Analytic conductor: \(15.4551\)
Root analytic conductor: \(15.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (2283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3328,\ (0:\ ),\ 0.170 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.424725465 + 2.041310009i\)
\(L(\frac12)\) \(\approx\) \(2.424725465 + 2.041310009i\)
\(L(1)\) \(\approx\) \(1.645404823 + 0.6251367461i\)
\(L(1)\) \(\approx\) \(1.645404823 + 0.6251367461i\)

Euler product

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

−18.72827945144890984569180588951, −18.04626799668757477756298315196, −17.38520205593566377219456008680, −16.78613608201184643669101547695, −16.01243196338876000708566837997, −14.91728685721757205275824692712, −14.233167495277709004880571080157, −14.00013143325232584061021881920, −13.23208228720356322593244929393, −12.70791901727520188333134009649, −11.66724668167439794906997196306, −11.04414950912300633446733637574, −9.96988184841625586158447493369, −9.38765506244305754915679501879, −9.09755421689476932837371007737, −7.63744364869129309116726888306, −7.457334765106294782606644631594, −6.58868852652331467049238968963, −5.94425678724637427788615512920, −4.998638061585454218366894262167, −3.719302867944656884028152750541, −3.404688674535120320280671957236, −2.36304910237080296284207297368, −1.50327235236948446367080189856, −0.877155359060857976353832460448, 1.31343575375429651264724797354, 2.0075832256506953801826988964, 2.83972224794563987424924036864, 3.55275599514326967840495535856, 4.51771002470792611843385057112, 5.30067600510798521402340487979, 5.912936222744897329717518265919, 6.78426200665742033726956592597, 7.74353905864923057711413108866, 8.846634575871174345514404948620, 8.992885207767992588409541923678, 9.61141933172665303046843690816, 10.43814558055569556268250050872, 11.11971968421504073287214124211, 12.276298221034443796269232972609, 12.6783026322407067107455300977, 13.626435947932061117798482106503, 14.199425595334106914169320433463, 14.85116910815165516781625986984, 15.38622852400465852841217789363, 16.315319111959198508849249318838, 16.79751606894261937375149812858, 17.58239300783234463123981652646, 18.37107620189481861521555818081, 19.14367870934581528390646701760

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