Properties

Label 2-2475-11.10-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9106919591\)
\(L(\frac12)\) \(\approx\) \(0.9106919591\)
\(L(1)\) 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

−9.255446755635192883616413321123, −8.665890775692716984870400346682, −7.86787089806435487365283782211, −6.64864046221455927308715424955, −5.94928391167267172541956313069, −4.95013718549775742299814285568, −4.12775050035510841839462947434, −3.12705762268618250485070417587, −2.29292202697258242983783521447, −1.66272384420624847595547089639, 0.58782236177049019704470515032, 2.53491261925341392823887334934, 3.59593173238698278212631421531, 4.76344684937300545934072862640, 5.24750873227289428694305017670, 6.07882341212895919678951994991, 7.11162967738839176791672166010, 7.42725775038231072163251504412, 8.008697454278127680228133237409, 8.796382487019720156064712316084

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