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Properties

Label 2-2475-11.10-c0-0-4
Degree $2$
Conductor $2475$
Sign $-1$
Analytic cond. $1.23518$
Root an. cond. $1.11138$
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  − 1.41i·2-s − 1.00·4-s − 1.41i·7-s + 11-s − 1.41i·13-s − 2.00·14-s − 0.999·16-s + 1.41i·17-s − 1.41i·22-s − 2.00·26-s + 1.41i·28-s + 1.41i·32-s + 2.00·34-s − 1.41i·43-s − 1.00·44-s + ⋯
L(s)  = 1  − 1.41i·2-s − 1.00·4-s − 1.41i·7-s + 11-s − 1.41i·13-s − 2.00·14-s − 0.999·16-s + 1.41i·17-s − 1.41i·22-s − 2.00·26-s + 1.41i·28-s + 1.41i·32-s + 2.00·34-s − 1.41i·43-s − 1.00·44-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(1.23518\)
Root analytic conductor: \(1.11138\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.184399837\)
\(L(\frac12)\) \(\approx\) \(1.184399837\)
\(L(1)\) 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 \( 1 - T \)
good2 \( 1 + 1.41iT - T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.41iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.41iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + 2T + T^{2} \)
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

−8.897213897461852314347870728101, −8.139626582578233471673799587764, −7.21442510577589526809801131629, −6.50316908836612358804794293084, −5.43950039264207885793310037888, −4.14989250529904364092095819238, −3.86423390662753462415709507789, −2.98543465640059473653356308600, −1.70428658917981024818608436446, −0.830067582061538016029864266451, 1.85598143194806943782795038501, 2.89133282975621820332282598464, 4.31087894259796945565522230331, 4.98190229141573149935883669302, 5.82854231331736859325744871499, 6.47194816248568494855302905191, 7.01917485879440151615211308252, 7.87279740722943014216247263140, 8.793860599828825712313843901958, 9.148943117436535000733471307183

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