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Properties

Label 2-3525-141.8-c0-0-1
Degree $2$
Conductor $3525$
Sign $-0.587 - 0.808i$
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  + (0.680 − 1.56i)2-s + (−0.997 + 0.0682i)3-s + (−1.31 − 1.40i)4-s + (−0.572 + 1.61i)6-s + (−1.48 + 0.526i)8-s + (0.990 − 0.136i)9-s + (1.40 + 1.31i)12-s + (−0.0523 + 0.764i)16-s + (−0.256 − 0.315i)17-s + (0.461 − 1.64i)18-s + (−1.16 + 0.709i)19-s + (−0.730 − 1.68i)23-s + (1.44 − 0.626i)24-s + (−0.979 + 0.203i)27-s + (0.105 − 1.54i)31-s + (−0.232 − 0.120i)32-s + ⋯
L(s)  = 1  + (0.680 − 1.56i)2-s + (−0.997 + 0.0682i)3-s + (−1.31 − 1.40i)4-s + (−0.572 + 1.61i)6-s + (−1.48 + 0.526i)8-s + (0.990 − 0.136i)9-s + (1.40 + 1.31i)12-s + (−0.0523 + 0.764i)16-s + (−0.256 − 0.315i)17-s + (0.461 − 1.64i)18-s + (−1.16 + 0.709i)19-s + (−0.730 − 1.68i)23-s + (1.44 − 0.626i)24-s + (−0.979 + 0.203i)27-s + (0.105 − 1.54i)31-s + (−0.232 − 0.120i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.587 - 0.808i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (3251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ -0.587 - 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7316272823\)
\(L(\frac12)\) \(\approx\) \(0.7316272823\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.997 - 0.0682i)T \)
5 \( 1 \)
47 \( 1 + (0.816 + 0.576i)T \)
good2 \( 1 + (-0.680 + 1.56i)T + (-0.682 - 0.730i)T^{2} \)
7 \( 1 + (-0.917 - 0.398i)T^{2} \)
11 \( 1 + (-0.203 - 0.979i)T^{2} \)
13 \( 1 + (0.854 - 0.519i)T^{2} \)
17 \( 1 + (0.256 + 0.315i)T + (-0.203 + 0.979i)T^{2} \)
19 \( 1 + (1.16 - 0.709i)T + (0.460 - 0.887i)T^{2} \)
23 \( 1 + (0.730 + 1.68i)T + (-0.682 + 0.730i)T^{2} \)
29 \( 1 + (-0.854 - 0.519i)T^{2} \)
31 \( 1 + (-0.105 + 1.54i)T + (-0.990 - 0.136i)T^{2} \)
37 \( 1 + (-0.334 + 0.942i)T^{2} \)
41 \( 1 + (0.775 + 0.631i)T^{2} \)
43 \( 1 + (-0.0682 + 0.997i)T^{2} \)
53 \( 1 + (1.86 + 0.663i)T + (0.775 + 0.631i)T^{2} \)
59 \( 1 + (0.0682 + 0.997i)T^{2} \)
61 \( 1 + (1.11 - 1.57i)T + (-0.334 - 0.942i)T^{2} \)
67 \( 1 + (-0.917 + 0.398i)T^{2} \)
71 \( 1 + (-0.682 + 0.730i)T^{2} \)
73 \( 1 + (0.962 + 0.269i)T^{2} \)
79 \( 1 + (-0.713 + 1.37i)T + (-0.576 - 0.816i)T^{2} \)
83 \( 1 + (-0.422 + 0.519i)T + (-0.203 - 0.979i)T^{2} \)
89 \( 1 + (-0.460 - 0.887i)T^{2} \)
97 \( 1 + (-0.990 + 0.136i)T^{2} \)
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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

−8.471260564204569650235562367767, −7.47731507838751249802270372755, −6.31923496550165551245780445227, −5.94745267055893115436874247968, −4.82092390761640527332427952097, −4.40450106486270848559143351792, −3.68419126716932349917444683656, −2.48865234267415559545663040045, −1.72690579057829960861612554959, −0.38022995916151234034253780931, 1.71226612235880071380909733123, 3.40471758510654415553669711607, 4.29538298716939485998082463393, 4.91330176723526158890058213295, 5.56884536254395149021699484544, 6.34495681992927787953091160476, 6.70567692277009389761607278459, 7.52901763951359964597842461031, 8.121318431173072118754195082722, 8.997546673177830281840359910170

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