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Properties

Label 2-1575-105.62-c0-0-8
Degree $2$
Conductor $1575$
Sign $0.465 + 0.884i$
Analytic cond. $0.786027$
Root an. cond. $0.886581$
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  + (−0.366 + 0.366i)2-s + 0.732i·4-s + (−0.707 − 0.707i)7-s + (−0.633 − 0.633i)8-s − 1.93i·11-s + 0.517·14-s − 0.267·16-s + (0.707 + 0.707i)22-s + (−1.36 − 1.36i)23-s + (0.517 − 0.517i)28-s − 0.517·29-s + (0.732 − 0.732i)32-s + (−0.707 − 0.707i)37-s + (0.707 − 0.707i)43-s + 1.41·44-s + ⋯
L(s)  = 1  + (−0.366 + 0.366i)2-s + 0.732i·4-s + (−0.707 − 0.707i)7-s + (−0.633 − 0.633i)8-s − 1.93i·11-s + 0.517·14-s − 0.267·16-s + (0.707 + 0.707i)22-s + (−1.36 − 1.36i)23-s + (0.517 − 0.517i)28-s − 0.517·29-s + (0.732 − 0.732i)32-s + (−0.707 − 0.707i)37-s + (0.707 − 0.707i)43-s + 1.41·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $0.465 + 0.884i$
Analytic conductor: \(0.786027\)
Root analytic conductor: \(0.886581\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1575,\ (\ :0),\ 0.465 + 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5824026354\)
\(L(\frac12)\) \(\approx\) \(0.5824026354\)
\(L(1)\) 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 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
11 \( 1 + 1.93iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
29 \( 1 + 0.517T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
71 \( 1 + 0.517iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + iT - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.227919670187391511678118573687, −8.622751879436082587289266151492, −7.967846832131971277377909073181, −7.13714814090795498483808518038, −6.34083352123632998494843505776, −5.70693674536575711845824443710, −4.12527080358336090050038417874, −3.57696924572056915400755650890, −2.61439408513177880902926591006, −0.50012489465889861143531675321, 1.71537143237607301999522158749, 2.41625133245544122439162071957, 3.75979719155097544897492757450, 4.92206129013777442742288408623, 5.65399203099012365497566509602, 6.50927997815118777455963151415, 7.32239974841637978429794039386, 8.329383115485110878137850860668, 9.346142775819755846682483830049, 9.746390656935353388900641372813

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