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Properties

Label 2-3775-755.754-c0-0-4
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  − 1.80i·2-s − 2.24·4-s + 2.24i·8-s − 9-s + 1.24·11-s + 1.80·16-s + 1.24i·17-s + 1.80i·18-s + 1.80·19-s − 2.24i·22-s + 1.80·29-s − 0.445·31-s − 1.00i·32-s + 2.24·34-s + 2.24·36-s + 1.24i·37-s + ⋯
L(s)  = 1  − 1.80i·2-s − 2.24·4-s + 2.24i·8-s − 9-s + 1.24·11-s + 1.80·16-s + 1.24i·17-s + 1.80i·18-s + 1.80·19-s − 2.24i·22-s + 1.80·29-s − 0.445·31-s − 1.00i·32-s + 2.24·34-s + 2.24·36-s + 1.24i·37-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

\(L(\frac{1}{2})\) \(\approx\) \(1.131275736\)
\(L(\frac12)\) \(\approx\) \(1.131275736\)
\(L(1)\) 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 \( 1 + 1.80iT - T^{2} \)
3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 1.24T + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - 1.24iT - T^{2} \)
19 \( 1 - 1.80T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - 1.80T + T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 - 1.24iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 0.445iT - T^{2} \)
47 \( 1 + 1.80iT - T^{2} \)
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 \( 1 + 1.80iT - T^{2} \)
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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

−8.528623042965475047282541786150, −8.306050369882130442259142763752, −6.93550079356539616570030589315, −6.03243372361690106619167776685, −5.15754399258540317658331868925, −4.34564729275751148052026539636, −3.42821844728194918783113049282, −3.03504958908763192016373456249, −1.84706038795290864350960804330, −1.00647529105791423415401509010, 0.915412614196979305723240565686, 2.83060586529036865115992022655, 3.76609012320083710080404005802, 4.78516461293196580752167749143, 5.32631335636157548207998012024, 6.09148723622790704920903722893, 6.69937451553300069133884342141, 7.39795983450216079240418593530, 7.981061762550776991611188714464, 8.840480436309148009892933066923

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