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Properties

Label 1-3512-3512.877-r1-0-0
Degree $1$
Conductor $3512$
Sign $1$
Analytic cond. $377.416$
Root an. cond. $377.416$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 7-s + 9-s − 11-s − 13-s − 15-s − 17-s − 19-s + 21-s − 23-s + 25-s + 27-s − 29-s − 31-s − 33-s − 35-s + 37-s − 39-s − 41-s + 43-s − 45-s − 47-s + 49-s − 51-s − 53-s + 55-s + ⋯
L(s)  = 1  + 3-s − 5-s + 7-s + 9-s − 11-s − 13-s − 15-s − 17-s − 19-s + 21-s − 23-s + 25-s + 27-s − 29-s − 31-s − 33-s − 35-s + 37-s − 39-s − 41-s + 43-s − 45-s − 47-s + 49-s − 51-s − 53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3512 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3512 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3512\)    =    \(2^{3} \cdot 439\)
Sign: $1$
Analytic conductor: \(377.416\)
Root analytic conductor: \(377.416\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3512} (877, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 3512,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.567329627\)
\(L(\frac12)\) \(\approx\) \(1.567329627\)
\(L(1)\) \(\approx\) \(1.060236166\)
\(L(1)\) \(\approx\) \(1.060236166\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
439 \( 1 \)
good3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 - T \)
97 \( 1 - T \)
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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

−18.54011726053752702052026797923, −18.17384147137676028834138969398, −17.20603333256830886487042811174, −16.36684308764669087941508523451, −15.54125175794724145126664165975, −15.12342740478806687797402777121, −14.56543456731608367213916931289, −13.90003544287572008083053635475, −12.81328963341850673002506827999, −12.643654904042258149293142123063, −11.46160713509087748421705408834, −10.97712624318472595007099646724, −10.16054167413613108739850656219, −9.302658981596017606559891109916, −8.447359971573188280934582022413, −8.01396306373899312555829988789, −7.46863988101539586434066134594, −6.77004074144847274377295152358, −5.45837179927261407872275360130, −4.55410157786780995125510311263, −4.21943023070718914613983356330, −3.23800955106593291256326681853, −2.28042807182069228221129810050, −1.866797821377750070985461016975, −0.40448449818339222164583885554, 0.40448449818339222164583885554, 1.866797821377750070985461016975, 2.28042807182069228221129810050, 3.23800955106593291256326681853, 4.21943023070718914613983356330, 4.55410157786780995125510311263, 5.45837179927261407872275360130, 6.77004074144847274377295152358, 7.46863988101539586434066134594, 8.01396306373899312555829988789, 8.447359971573188280934582022413, 9.302658981596017606559891109916, 10.16054167413613108739850656219, 10.97712624318472595007099646724, 11.46160713509087748421705408834, 12.643654904042258149293142123063, 12.81328963341850673002506827999, 13.90003544287572008083053635475, 14.56543456731608367213916931289, 15.12342740478806687797402777121, 15.54125175794724145126664165975, 16.36684308764669087941508523451, 17.20603333256830886487042811174, 18.17384147137676028834138969398, 18.54011726053752702052026797923

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