Properties

Label 2-3775-151.150-c0-0-16
Degree $2$
Conductor $3775$
Sign $i$
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  + 0.618·2-s − i·3-s − 0.618·4-s − 0.618i·6-s + 0.618i·7-s − 8-s + 0.618i·12-s − 1.61i·13-s + 0.381i·14-s + 17-s + 19-s + 0.618·21-s + i·23-s + i·24-s − 1.00i·26-s − i·27-s + ⋯
L(s)  = 1  + 0.618·2-s − i·3-s − 0.618·4-s − 0.618i·6-s + 0.618i·7-s − 8-s + 0.618i·12-s − 1.61i·13-s + 0.381i·14-s + 17-s + 19-s + 0.618·21-s + i·23-s + i·24-s − 1.00i·26-s − i·27-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3775\)    =    \(5^{2} \cdot 151\)
Sign: $i$
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: \((2,\ 3775,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.420785371\)
\(L(\frac12)\) \(\approx\) \(1.420785371\)
\(L(1)\) 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 \( 1 - iT \)
good2 \( 1 - 0.618T + T^{2} \)
3 \( 1 + iT - T^{2} \)
7 \( 1 - 0.618iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 1.61iT - T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 0.618T + T^{2} \)
37 \( 1 + 1.61T + T^{2} \)
41 \( 1 + 1.61iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + 1.61T + T^{2} \)
53 \( 1 + 1.61iT - T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - 0.618iT - T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + 1.61iT - T^{2} \)
79 \( 1 + 1.61iT - T^{2} \)
83 \( 1 + 0.618iT - T^{2} \)
89 \( 1 - 0.618iT - T^{2} \)
97 \( 1 - 0.618T + T^{2} \)
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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

−8.356152309743178580180421369429, −7.69125642542700662968130922583, −7.13864064928261648864673582632, −6.00655345452023729324485065999, −5.48385502672489785889499914210, −5.03528887251625743434008899190, −3.59967666108953883101249996229, −3.21543545390719936349972459744, −1.98133852457180313363502108659, −0.76227224747650925565772199920, 1.33951671390907238990404458676, 2.93050437443074600234791159263, 3.79542220480825197219911643365, 4.22890677732717585946307372905, 4.92352573091255078295213086370, 5.55742618288086807949974596595, 6.59729945288733875152045347676, 7.27492893544287433859077226624, 8.286497488199239776559521721664, 9.064694774996407402087407006812

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