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Properties

Label 2-3525-705.422-c0-0-21
Degree $2$
Conductor $3525$
Sign $-0.649 + 0.760i$
Analytic cond. $1.75920$
Root an. cond. $1.32634$
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  + (1.29 − 1.29i)2-s + (−0.838 − 0.544i)3-s − 2.33i·4-s + (−1.78 + 0.379i)6-s + (1.40 + 1.40i)7-s + (−1.72 − 1.72i)8-s + (0.406 + 0.913i)9-s + (−1.27 + 1.96i)12-s + 3.63·14-s − 2.12·16-s + (0.946 − 0.946i)17-s + (1.70 + 0.654i)18-s + (−0.413 − 1.94i)21-s + (0.508 + 2.39i)24-s + (0.156 − 0.987i)27-s + (3.28 − 3.28i)28-s + ⋯
L(s)  = 1  + (1.29 − 1.29i)2-s + (−0.838 − 0.544i)3-s − 2.33i·4-s + (−1.78 + 0.379i)6-s + (1.40 + 1.40i)7-s + (−1.72 − 1.72i)8-s + (0.406 + 0.913i)9-s + (−1.27 + 1.96i)12-s + 3.63·14-s − 2.12·16-s + (0.946 − 0.946i)17-s + (1.70 + 0.654i)18-s + (−0.413 − 1.94i)21-s + (0.508 + 2.39i)24-s + (0.156 − 0.987i)27-s + (3.28 − 3.28i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.649 + 0.760i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1832, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.649 + 0.760i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.283859497\)
\(L(\frac12)\) \(\approx\) \(2.283859497\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.838 + 0.544i)T \)
5 \( 1 \)
47 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-1.29 + 1.29i)T - iT^{2} \)
7 \( 1 + (-1.40 - 1.40i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-0.946 + 0.946i)T - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1.34 + 1.34i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
53 \( 1 + (1.14 + 1.14i)T + iT^{2} \)
59 \( 1 - 1.98T + T^{2} \)
61 \( 1 - 1.61T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 0.415iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 0.618iT - T^{2} \)
83 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
89 \( 1 + 1.17T + T^{2} \)
97 \( 1 + (-0.831 - 0.831i)T + iT^{2} \)
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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

−8.514408610822239776685190381509, −7.68552961602923353387255043004, −6.66115728539909358454740509562, −5.70161065339234812403759055222, −5.24470710086684080859059331987, −4.94082313213898890414046711643, −3.85037239211913720974596731165, −2.65319215468589828012299581733, −2.03411897163849376662763608578, −1.18777318081906914367997130618, 1.42345122762625830126732431736, 3.39114889018203210584666702535, 3.97929357864573120254978791078, 4.62714735365761040857362583080, 5.20613634701492257389946695209, 5.81907685444299811850278628817, 6.80954068081966609872261986697, 7.15917044094078693370545079548, 8.108209161849851711214295661082, 8.458165117209200072496745297214

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