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Properties

Label 2-3525-705.422-c0-0-28
Degree $2$
Conductor $3525$
Sign $-0.688 - 0.725i$
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  + (1.38 − 1.38i)2-s + (−0.629 + 0.777i)3-s − 2.82i·4-s + (0.204 + 1.94i)6-s + (−1.05 − 1.05i)7-s + (−2.52 − 2.52i)8-s + (−0.207 − 0.978i)9-s + (2.19 + 1.77i)12-s − 2.90·14-s − 4.16·16-s + (−1.29 + 1.29i)17-s + (−1.64 − 1.06i)18-s + (1.47 − 0.155i)21-s + (3.55 − 0.373i)24-s + (0.891 + 0.453i)27-s + (−2.97 + 2.97i)28-s + ⋯
L(s)  = 1  + (1.38 − 1.38i)2-s + (−0.629 + 0.777i)3-s − 2.82i·4-s + (0.204 + 1.94i)6-s + (−1.05 − 1.05i)7-s + (−2.52 − 2.52i)8-s + (−0.207 − 0.978i)9-s + (2.19 + 1.77i)12-s − 2.90·14-s − 4.16·16-s + (−1.29 + 1.29i)17-s + (−1.64 − 1.06i)18-s + (1.47 − 0.155i)21-s + (3.55 − 0.373i)24-s + (0.891 + 0.453i)27-s + (−2.97 + 2.97i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.688 - 0.725i$
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: \((2,\ 3525,\ (\ :0),\ -0.688 - 0.725i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.043619302\)
\(L(\frac12)\) \(\approx\) \(1.043619302\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.629 - 0.777i)T \)
5 \( 1 \)
47 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 + (-1.38 + 1.38i)T - iT^{2} \)
7 \( 1 + (1.05 + 1.05i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (1.29 - 1.29i)T - iT^{2} \)
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 \( 1 - 1.48T + T^{2} \)
61 \( 1 + 0.618T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + 1.98iT - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + 1.61iT - T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
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

−8.751900625842805631362285224193, −6.99509030966287196687970372473, −6.42955116188743198413271302879, −5.81564417588591367816189026428, −4.95261816647147514329899429172, −4.09516658064534882051037941509, −3.83409954741234224987122275500, −3.01016300364867463760458848937, −1.79557409215880849464592034112, −0.38152149203968294188715136713, 2.44036123422099809906781401298, 2.91942013351544269873072271787, 4.13745809120609334781982342546, 5.02786903625598304735411213819, 5.56226084196349043045288685494, 6.24790198084669801123084359577, 6.83095958400385405567468412925, 7.20768080598635896841521944789, 8.236539689354323469225645632113, 8.801232023314362070386393828737

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