Properties

Label 2-2475-33.32-c1-0-58
Degree $2$
Conductor $2475$
Sign $-0.174 + 0.984i$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 1.41i·7-s − 3·8-s + (3 − 1.41i)11-s + 2.82i·13-s − 1.41i·14-s − 16-s + 2·17-s − 4.24i·19-s + (3 − 1.41i)22-s + 2.82i·26-s + 1.41i·28-s − 6·29-s + 5·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.5·4-s − 0.534i·7-s − 1.06·8-s + (0.904 − 0.426i)11-s + 0.784i·13-s − 0.377i·14-s − 0.250·16-s + 0.485·17-s − 0.973i·19-s + (0.639 − 0.301i)22-s + 0.554i·26-s + 0.267i·28-s − 1.11·29-s + 0.883·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-0.174 + 0.984i$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (2276, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ -0.174 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.532598872\)
\(L(\frac12)\) \(\approx\) \(1.532598872\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 + (-3 + 1.41i)T \)
good2 \( 1 - T + 2T^{2} \)
7 \( 1 + 1.41iT - 7T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 4.24iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 1.41iT - 43T^{2} \)
47 \( 1 + 8.48iT - 47T^{2} \)
53 \( 1 + 4.24iT - 53T^{2} \)
59 \( 1 + 8.48iT - 59T^{2} \)
61 \( 1 + 8.48iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 - 7.07iT - 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−8.947429024077156820938337818459, −7.973457791729002794039568640565, −6.93781539653073318982600137940, −6.40950927484118579323489048208, −5.39146171517367558249847039900, −4.73872779296264986644860584122, −3.79573072757325229596260418073, −3.37323220293986808110865779505, −1.87650461439571599527840217535, −0.43328419087529532887566205777, 1.32763939073741645243102507143, 2.70317280267051485689814266796, 3.65043888187216562916594952017, 4.25223124601361947721854785288, 5.39788074511568265068825374384, 5.69690560716048593252894151112, 6.65693434760224074119764483491, 7.63862135316422984781261979684, 8.461342399726571342192181737833, 9.128653208233236826055818918593

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