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Properties

Label 2-3525-705.563-c0-0-19
Degree $2$
Conductor $3525$
Sign $-0.0746 + 0.997i$
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  + (0.437 + 0.437i)2-s + (−0.987 + 0.156i)3-s − 0.618i·4-s + (−0.5 − 0.363i)6-s + (−1.34 + 1.34i)7-s + (0.707 − 0.707i)8-s + (0.951 − 0.309i)9-s + (0.0966 + 0.610i)12-s − 1.17·14-s + (−1.14 − 1.14i)17-s + (0.550 + 0.280i)18-s + (1.11 − 1.53i)21-s + (−0.587 + 0.809i)24-s + (−0.891 + 0.453i)27-s + (0.831 + 0.831i)28-s + ⋯
L(s)  = 1  + (0.437 + 0.437i)2-s + (−0.987 + 0.156i)3-s − 0.618i·4-s + (−0.5 − 0.363i)6-s + (−1.34 + 1.34i)7-s + (0.707 − 0.707i)8-s + (0.951 − 0.309i)9-s + (0.0966 + 0.610i)12-s − 1.17·14-s + (−1.14 − 1.14i)17-s + (0.550 + 0.280i)18-s + (1.11 − 1.53i)21-s + (−0.587 + 0.809i)24-s + (−0.891 + 0.453i)27-s + (0.831 + 0.831i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0746 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0746 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.0746 + 0.997i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.0746 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4973338925\)
\(L(\frac12)\) \(\approx\) \(0.4973338925\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.987 - 0.156i)T \)
5 \( 1 \)
47 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-0.437 - 0.437i)T + iT^{2} \)
7 \( 1 + (1.34 - 1.34i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1.14 + 1.14i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.831 + 0.831i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (-0.437 + 0.437i)T - iT^{2} \)
59 \( 1 + 1.90T + T^{2} \)
61 \( 1 + 0.618T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.17iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.61iT - T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
89 \( 1 + 1.90T + T^{2} \)
97 \( 1 + (-1.34 + 1.34i)T - iT^{2} \)
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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

−8.811974475865775291570094634337, −7.41864305519676240879780417883, −6.69262603225285408119449584563, −6.24946618508265195684476112200, −5.63548178591960023893038667366, −4.99412403443837735064562083391, −4.22567724334287319270249238771, −3.06230321976909368927058544288, −1.96544144524055926533076938016, −0.28829615055330846065024616121, 1.36216423164237213625758795417, 2.67886019703285457449381449958, 3.70797964071447553541375151094, 4.20511082543203267845484013424, 4.90964257005752466830257878164, 6.25746723100800390826334333222, 6.48638029135014633715609328368, 7.39340953101794298131777404586, 7.930892737786354287367310628493, 9.053583595370399289396735216166

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