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Properties

Label 2-3775-151.150-c0-0-20
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 no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s − 4-s − 2i·7-s − 11-s + i·12-s − i·13-s + 16-s + 19-s − 2·21-s − i·23-s − i·27-s + 2i·28-s + 29-s − 31-s + i·33-s + ⋯
L(s)  = 1  − i·3-s − 4-s − 2i·7-s − 11-s + i·12-s − i·13-s + 16-s + 19-s − 2·21-s − i·23-s − i·27-s + 2i·28-s + 29-s − 31-s + i·33-s + ⋯

Functional equation

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

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: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3775,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8435708325\)
\(L(\frac12)\) \(\approx\) \(0.8435708325\)
\(L(1)\) 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 \( 1 + T^{2} \)
3 \( 1 + iT - T^{2} \)
7 \( 1 + 2iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−7.944699285361934814498419044941, −7.68382408588969629534154056928, −7.11276362940288740695077900112, −6.19073900482827823323904484501, −5.19983717399114859790597186899, −4.51494026800878213961268668497, −3.71920323583216818194869987650, −2.79394457645728698115400824278, −1.24006977328386617471232795097, −0.56362048584752421516337063825, 1.80489770333143527872182114322, 2.93030459474745716641216190305, 3.69540554842032483584573017146, 4.67368616247789027810872882097, 5.36267372326339950231055015161, 5.52500465361223746700874343235, 6.77815614046013358688306501164, 7.88735517749603645273196182660, 8.553432982678387484309584107362, 9.165056672024541394751733806096

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