Properties

Label 2-1575-21.20-c1-0-24
Degree $2$
Conductor $1575$
Sign $0.692 + 0.721i$
Analytic cond. $12.5764$
Root an. cond. $3.54632$
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  − 1.41i·2-s + (−0.5 + 2.59i)7-s − 2.82i·8-s − 1.41i·11-s + 5.19i·13-s + (3.67 + 0.707i)14-s − 4.00·16-s + 7.34·17-s − 5.19i·19-s − 2.00·22-s + 7.07i·23-s + 7.34·26-s − 1.41i·29-s + 5.19i·31-s − 10.3i·34-s + ⋯
L(s)  = 1  − 0.999i·2-s + (−0.188 + 0.981i)7-s − 0.999i·8-s − 0.426i·11-s + 1.44i·13-s + (0.981 + 0.188i)14-s − 1.00·16-s + 1.78·17-s − 1.19i·19-s − 0.426·22-s + 1.47i·23-s + 1.44·26-s − 0.262i·29-s + 0.933i·31-s − 1.78i·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.692 + 0.721i$
Analytic conductor: \(12.5764\)
Root analytic conductor: \(3.54632\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :1/2),\ 0.692 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.965492438\)
\(L(\frac12)\) \(\approx\) \(1.965492438\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (0.5 - 2.59i)T \)
good2 \( 1 + 1.41iT - 2T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 - 7.34T + 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 - 7.07iT - 23T^{2} \)
29 \( 1 + 1.41iT - 29T^{2} \)
31 \( 1 - 5.19iT - 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 - 7.34T + 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 - 7.34T + 59T^{2} \)
61 \( 1 + 5.19iT - 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 7.34T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 + 5.19iT - 97T^{2} \)
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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

−9.428637791516586588657662983583, −8.912405007858579811000494076777, −7.73319924978320057180608151471, −6.88939710659647541474868346019, −6.02974905111421388351410693684, −5.15916932600175843795756897853, −3.93694846463285587294594276301, −3.09910890846450230163799822959, −2.24737770046375000083303385871, −1.13630848281710471679405501188, 0.940149249540689580617938651699, 2.56793036557892588628104398178, 3.65008215603560782964561010883, 4.71816893428223097363821351136, 5.75473952560769904379158503179, 6.17632868371731035056422726068, 7.37883807629813188162529688174, 7.70116694471963126751880835765, 8.299939476372337503160887164242, 9.541808900039462334915150196969

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