Properties

Label 2-3525-141.131-c0-0-1
Degree $2$
Conductor $3525$
Sign $0.929 + 0.369i$
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  + (1.88 − 0.391i)2-s + (−0.730 + 0.682i)3-s + (2.48 − 1.07i)4-s + (−1.11 + 1.57i)6-s + (2.68 − 1.89i)8-s + (0.0682 − 0.997i)9-s + (−1.07 + 2.48i)12-s + (2.47 − 2.65i)16-s + (−1.46 − 0.519i)17-s + (−0.262 − 1.90i)18-s + (1.76 + 0.494i)19-s + (0.398 + 0.0827i)23-s + (−0.669 + 3.22i)24-s + (0.631 + 0.775i)27-s + (−0.457 + 0.489i)31-s + (1.92 − 3.15i)32-s + ⋯
L(s)  = 1  + (1.88 − 0.391i)2-s + (−0.730 + 0.682i)3-s + (2.48 − 1.07i)4-s + (−1.11 + 1.57i)6-s + (2.68 − 1.89i)8-s + (0.0682 − 0.997i)9-s + (−1.07 + 2.48i)12-s + (2.47 − 2.65i)16-s + (−1.46 − 0.519i)17-s + (−0.262 − 1.90i)18-s + (1.76 + 0.494i)19-s + (0.398 + 0.0827i)23-s + (−0.669 + 3.22i)24-s + (0.631 + 0.775i)27-s + (−0.457 + 0.489i)31-s + (1.92 − 3.15i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.929 + 0.369i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (2951, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.929 + 0.369i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.339421124\)
\(L(\frac12)\) \(\approx\) \(3.339421124\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.730 - 0.682i)T \)
5 \( 1 \)
47 \( 1 + (0.887 + 0.460i)T \)
good2 \( 1 + (-1.88 + 0.391i)T + (0.917 - 0.398i)T^{2} \)
7 \( 1 + (0.203 + 0.979i)T^{2} \)
11 \( 1 + (0.775 - 0.631i)T^{2} \)
13 \( 1 + (0.962 + 0.269i)T^{2} \)
17 \( 1 + (1.46 + 0.519i)T + (0.775 + 0.631i)T^{2} \)
19 \( 1 + (-1.76 - 0.494i)T + (0.854 + 0.519i)T^{2} \)
23 \( 1 + (-0.398 - 0.0827i)T + (0.917 + 0.398i)T^{2} \)
29 \( 1 + (-0.962 + 0.269i)T^{2} \)
31 \( 1 + (0.457 - 0.489i)T + (-0.0682 - 0.997i)T^{2} \)
37 \( 1 + (-0.576 + 0.816i)T^{2} \)
41 \( 1 + (0.334 + 0.942i)T^{2} \)
43 \( 1 + (0.682 - 0.730i)T^{2} \)
53 \( 1 + (-0.111 - 0.0787i)T + (0.334 + 0.942i)T^{2} \)
59 \( 1 + (-0.682 - 0.730i)T^{2} \)
61 \( 1 + (0.911 - 1.75i)T + (-0.576 - 0.816i)T^{2} \)
67 \( 1 + (0.203 - 0.979i)T^{2} \)
71 \( 1 + (0.917 + 0.398i)T^{2} \)
73 \( 1 + (-0.990 + 0.136i)T^{2} \)
79 \( 1 + (-0.572 - 0.347i)T + (0.460 + 0.887i)T^{2} \)
83 \( 1 + (1.08 - 0.386i)T + (0.775 - 0.631i)T^{2} \)
89 \( 1 + (-0.854 + 0.519i)T^{2} \)
97 \( 1 + (-0.0682 + 0.997i)T^{2} \)
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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

−8.924609868700771895039462693559, −7.47652154895210747505169145380, −6.83319298917301377308212355866, −6.18617273165154566660890361306, −5.29467953405950355202406412703, −5.02143544600188493601697054922, −4.13180556033075321093927664415, −3.46293804511208225499954476570, −2.65239077771426348287083405782, −1.36372209337494026140227324757, 1.59983690087475078512271061558, 2.56147226166816236979546493963, 3.43430882855498225422858555338, 4.54174711432677563889727630952, 4.95846949477740054686363876679, 5.81339280838774454824969510211, 6.33907985813749284822173164594, 7.06663824429226091888645477279, 7.53117745388159816672271059944, 8.370109716393518159774398020511

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