Properties

Label 1-3755-3755.3754-r1-0-0
Degree $1$
Conductor $3755$
Sign $1$
Analytic cond. $403.530$
Root an. cond. $403.530$
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  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s − 13-s − 14-s + 16-s + 17-s − 18-s + 19-s + 21-s + 22-s − 23-s − 24-s + 26-s + 27-s + 28-s − 29-s − 31-s − 32-s − 33-s − 34-s + ⋯
L(s)  = 1  − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 11-s + 12-s − 13-s − 14-s + 16-s + 17-s − 18-s + 19-s + 21-s + 22-s − 23-s − 24-s + 26-s + 27-s + 28-s − 29-s − 31-s − 32-s − 33-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3755 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3755 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3755\)    =    \(5 \cdot 751\)
Sign: $1$
Analytic conductor: \(403.530\)
Root analytic conductor: \(403.530\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3755} (3754, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 3755,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.103737798\)
\(L(\frac12)\) \(\approx\) \(2.103737798\)
\(L(1)\) \(\approx\) \(1.025356520\)
\(L(1)\) \(\approx\) \(1.025356520\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
751 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + T \)
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

−18.43248933007898422066992909322, −18.00249631940242784301176686620, −17.170283860353683575016730076913, −16.37050537012173679625791317307, −15.759493680743898008896693365741, −14.97922555673099784292687224278, −14.53191410165431459075925243148, −13.841189933601506409216031804266, −12.83734075625285061193132313639, −12.12944479899646338632030362389, −11.43192892017615343507623152111, −10.53958900633374892428097807268, −9.87031975306239148043283421053, −9.49481153654573386272723026148, −8.38101064705610455770548836753, −8.03767774744819427538576737759, −7.46233848185126912029871423871, −6.878594833520442395599183880088, −5.4495707922046837733962042145, −5.06857163346471286496414403665, −3.69373156871352042576317303795, −3.07008786546060047038267634708, −2.02742410321029057630418685920, −1.76127024745092412083444717632, −0.55703438878057944214003009524, 0.55703438878057944214003009524, 1.76127024745092412083444717632, 2.02742410321029057630418685920, 3.07008786546060047038267634708, 3.69373156871352042576317303795, 5.06857163346471286496414403665, 5.4495707922046837733962042145, 6.878594833520442395599183880088, 7.46233848185126912029871423871, 8.03767774744819427538576737759, 8.38101064705610455770548836753, 9.49481153654573386272723026148, 9.87031975306239148043283421053, 10.53958900633374892428097807268, 11.43192892017615343507623152111, 12.12944479899646338632030362389, 12.83734075625285061193132313639, 13.841189933601506409216031804266, 14.53191410165431459075925243148, 14.97922555673099784292687224278, 15.759493680743898008896693365741, 16.37050537012173679625791317307, 17.170283860353683575016730076913, 18.00249631940242784301176686620, 18.43248933007898422066992909322

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