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Properties

Label 2-3775-151.150-c0-0-18
Degree $2$
Conductor $3775$
Sign $1$
Analytic cond. $1.88397$
Root an. cond. $1.37257$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.80·2-s + 2.24·4-s + 2.24·8-s + 9-s + 1.24·11-s + 1.80·16-s − 1.24·17-s + 1.80·18-s − 1.80·19-s + 2.24·22-s − 1.80·29-s − 0.445·31-s + 1.00·32-s − 2.24·34-s + 2.24·36-s − 1.24·37-s − 3.24·38-s + 0.445·43-s + 2.80·44-s + 1.80·47-s + 49-s − 3.24·58-s − 0.445·59-s − 0.801·62-s − 2.80·68-s + 2.24·72-s − 2.24·74-s + ⋯
L(s)  = 1  + 1.80·2-s + 2.24·4-s + 2.24·8-s + 9-s + 1.24·11-s + 1.80·16-s − 1.24·17-s + 1.80·18-s − 1.80·19-s + 2.24·22-s − 1.80·29-s − 0.445·31-s + 1.00·32-s − 2.24·34-s + 2.24·36-s − 1.24·37-s − 3.24·38-s + 0.445·43-s + 2.80·44-s + 1.80·47-s + 49-s − 3.24·58-s − 0.445·59-s − 0.801·62-s − 2.80·68-s + 2.24·72-s − 2.24·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3775\)    =    \(5^{2} \cdot 151\)
Sign: $1$
Analytic conductor: \(1.88397\)
Root analytic conductor: \(1.37257\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3775} (301, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3775,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.042090602\)
\(L(\frac12)\) \(\approx\) \(4.042090602\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
151 \( 1 - T \)
good2 \( 1 - 1.80T + T^{2} \)
3 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.24T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.24T + T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.80T + T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 + 1.24T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 0.445T + T^{2} \)
47 \( 1 - 1.80T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 0.445T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 1.80T + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.766729020085522782349725988444, −7.44734835806918879809107050943, −6.92690939862875350444298030509, −6.35913577816417254508256053414, −5.66111816752167155735364636232, −4.66204769349471983720043349175, −4.04725519280319780344487350648, −3.73411336437972499935707719879, −2.31721536029407938427762244016, −1.73171358916206023895815435928, 1.73171358916206023895815435928, 2.31721536029407938427762244016, 3.73411336437972499935707719879, 4.04725519280319780344487350648, 4.66204769349471983720043349175, 5.66111816752167155735364636232, 6.35913577816417254508256053414, 6.92690939862875350444298030509, 7.44734835806918879809107050943, 8.766729020085522782349725988444

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