Properties

Label 2-3775-755.754-c0-0-13
Degree $2$
Conductor $3775$
Sign $0.447 - 0.894i$
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  − 1.61i·2-s + 3-s − 1.61·4-s − 1.61i·6-s − 1.61·7-s + i·8-s − 1.61·12-s − 0.618·13-s + 2.61i·14-s + i·17-s − 19-s − 1.61·21-s − 23-s + i·24-s + 1.00i·26-s − 27-s + ⋯
L(s)  = 1  − 1.61i·2-s + 3-s − 1.61·4-s − 1.61i·6-s − 1.61·7-s + i·8-s − 1.61·12-s − 0.618·13-s + 2.61i·14-s + i·17-s − 19-s − 1.61·21-s − 23-s + i·24-s + 1.00i·26-s − 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3775\)    =    \(5^{2} \cdot 151\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(1.88397\)
Root analytic conductor: \(1.37257\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3775} (3774, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3775,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01102138094\)
\(L(\frac12)\) \(\approx\) \(0.01102138094\)
\(L(1)\) 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 \( 1 + 1.61iT - T^{2} \)
3 \( 1 - T + T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 0.618T + T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.61T + T^{2} \)
37 \( 1 - 0.618iT - T^{2} \)
41 \( 1 + 0.618iT - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 - 0.618iT - T^{2} \)
53 \( 1 + 0.618T + T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - 1.61iT - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + 0.618T + T^{2} \)
79 \( 1 - 0.618iT - T^{2} \)
83 \( 1 + 1.61T + T^{2} \)
89 \( 1 + 1.61iT - T^{2} \)
97 \( 1 + 1.61iT - T^{2} \)
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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

−8.541902224138418274112229849525, −7.57301127838500333259863970590, −6.51660350482645867506539017041, −5.89170041239913611051688125495, −4.43793002373826629663241372317, −3.88131483052838497963865369062, −3.04960909541265295676369614560, −2.63657156767033294756524639026, −1.72890981498032702164640045429, −0.00495637426540444703759532887, 2.40547382148347986724129118386, 3.07154859658902465893959632514, 4.08066552089131055393636449750, 4.88150404557051602225106409148, 5.97098794823677318292118030232, 6.37070160712515077713136068285, 7.08728124190746137926912281404, 7.80193242245406633550302737328, 8.371334216364714775638282302619, 9.095598986110687240650685493114

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