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Properties

Label 1-3328-3328.51-r1-0-0
Degree $1$
Conductor $3328$
Sign $-0.757 + 0.653i$
Analytic cond. $357.643$
Root an. cond. $357.643$
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  + (0.956 − 0.290i)3-s + (−0.995 + 0.0980i)5-s + (0.555 − 0.831i)7-s + (0.831 − 0.555i)9-s + (−0.471 − 0.881i)11-s + (−0.923 + 0.382i)15-s + (−0.923 − 0.382i)17-s + (−0.634 − 0.773i)19-s + (0.290 − 0.956i)21-s + (−0.195 + 0.980i)23-s + (0.980 − 0.195i)25-s + (0.634 − 0.773i)27-s + (0.881 + 0.471i)29-s + (0.707 − 0.707i)31-s + (−0.707 − 0.707i)33-s + ⋯
L(s)  = 1  + (0.956 − 0.290i)3-s + (−0.995 + 0.0980i)5-s + (0.555 − 0.831i)7-s + (0.831 − 0.555i)9-s + (−0.471 − 0.881i)11-s + (−0.923 + 0.382i)15-s + (−0.923 − 0.382i)17-s + (−0.634 − 0.773i)19-s + (0.290 − 0.956i)21-s + (−0.195 + 0.980i)23-s + (0.980 − 0.195i)25-s + (0.634 − 0.773i)27-s + (0.881 + 0.471i)29-s + (0.707 − 0.707i)31-s + (−0.707 − 0.707i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $-0.757 + 0.653i$
Analytic conductor: \(357.643\)
Root analytic conductor: \(357.643\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3328,\ (1:\ ),\ -0.757 + 0.653i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3283362109 - 0.8833118227i\)
\(L(\frac12)\) \(\approx\) \(-0.3283362109 - 0.8833118227i\)
\(L(1)\) \(\approx\) \(1.045535196 - 0.4239830618i\)
\(L(1)\) \(\approx\) \(1.045535196 - 0.4239830618i\)

Euler product

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

−19.16970856134307052890137092884, −18.46337901416259743468430405221, −18.00451491293019262468523701915, −16.90701142981897507092944008650, −16.07525675649068579964689630886, −15.47776631584222292870857048340, −14.95074669003678734740961360607, −14.60149700578145074277449877142, −13.55585060225434404033978797083, −12.75251569859424745854555562541, −12.21142995558419783688668895431, −11.48882414047367539236445101223, −10.468346354739907163603184016696, −10.07139461681710710151395898471, −8.83953286014008168900222749770, −8.508358603184595491005933789820, −7.96653270389300783632853578030, −7.146766500780305755940435353122, −6.31216336057422025526024219226, −5.04600647930076045894815702144, −4.48156687292562110224240391421, −3.93818520056564884382348301082, −2.794025693281522423008426017936, −2.28768418101746443048360756131, −1.36171643522720912406892761413, 0.137862295970501804162222175239, 0.87853870573324034866077127902, 1.91102278024876331826576832501, 2.94094703842195305201692480946, 3.45600024055083959009495905648, 4.415411209914607726851147698643, 4.81842980215681369409469770305, 6.29225508077758728391568977179, 7.01876359724850743450522471994, 7.60487927910891530942110280568, 8.388397650482553302773458594765, 8.62988431754778076912199719337, 9.76760077313653624421085423619, 10.55835233980915135527006761085, 11.33935333130122593565581240621, 11.77453167266218734633391592451, 12.9775129265723720523903716126, 13.407663455107517631644563094615, 14.02199585236229916370360410273, 14.8066421561385557087750408444, 15.48764914176780915699921038022, 15.91739166819226822723661246198, 16.86186587947350373331887660818, 17.6936175709172887316486522792, 18.39066598146650067283115219007

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