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Properties

Label 2-3525-705.398-c0-0-1
Degree $2$
Conductor $3525$
Sign $0.329 - 0.944i$
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  + (1.97 + 0.202i)2-s + (−0.366 + 0.930i)3-s + (2.86 + 0.595i)4-s + (−0.911 + 1.75i)6-s + (3.63 + 1.15i)8-s + (−0.730 − 0.682i)9-s + (−1.60 + 2.44i)12-s + (4.25 + 1.84i)16-s + (0.660 − 0.113i)17-s + (−1.30 − 1.49i)18-s + (−1.93 + 0.266i)19-s + (−0.129 − 1.25i)23-s + (−2.40 + 2.95i)24-s + (0.903 − 0.429i)27-s + (−0.650 + 1.49i)31-s + (4.67 + 2.63i)32-s + ⋯
L(s)  = 1  + (1.97 + 0.202i)2-s + (−0.366 + 0.930i)3-s + (2.86 + 0.595i)4-s + (−0.911 + 1.75i)6-s + (3.63 + 1.15i)8-s + (−0.730 − 0.682i)9-s + (−1.60 + 2.44i)12-s + (4.25 + 1.84i)16-s + (0.660 − 0.113i)17-s + (−1.30 − 1.49i)18-s + (−1.93 + 0.266i)19-s + (−0.129 − 1.25i)23-s + (−2.40 + 2.95i)24-s + (0.903 − 0.429i)27-s + (−0.650 + 1.49i)31-s + (4.67 + 2.63i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.329 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.329 - 0.944i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (3218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.329 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.923695440\)
\(L(\frac12)\) \(\approx\) \(3.923695440\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.366 - 0.930i)T \)
5 \( 1 \)
47 \( 1 + (0.236 + 0.971i)T \)
good2 \( 1 + (-1.97 - 0.202i)T + (0.979 + 0.203i)T^{2} \)
7 \( 1 + (0.631 + 0.775i)T^{2} \)
11 \( 1 + (-0.334 + 0.942i)T^{2} \)
13 \( 1 + (0.136 + 0.990i)T^{2} \)
17 \( 1 + (-0.660 + 0.113i)T + (0.942 - 0.334i)T^{2} \)
19 \( 1 + (1.93 - 0.266i)T + (0.962 - 0.269i)T^{2} \)
23 \( 1 + (0.129 + 1.25i)T + (-0.979 + 0.203i)T^{2} \)
29 \( 1 + (0.990 + 0.136i)T^{2} \)
31 \( 1 + (0.650 - 1.49i)T + (-0.682 - 0.730i)T^{2} \)
37 \( 1 + (-0.887 - 0.460i)T^{2} \)
41 \( 1 + (-0.576 - 0.816i)T^{2} \)
43 \( 1 + (-0.398 + 0.917i)T^{2} \)
53 \( 1 + (0.412 + 1.30i)T + (-0.816 + 0.576i)T^{2} \)
59 \( 1 + (-0.917 + 0.398i)T^{2} \)
61 \( 1 + (-0.116 - 0.0709i)T + (0.460 + 0.887i)T^{2} \)
67 \( 1 + (0.631 - 0.775i)T^{2} \)
71 \( 1 + (-0.203 - 0.979i)T^{2} \)
73 \( 1 + (0.997 + 0.0682i)T^{2} \)
79 \( 1 + (-0.311 - 1.11i)T + (-0.854 + 0.519i)T^{2} \)
83 \( 1 + (0.906 + 0.156i)T + (0.942 + 0.334i)T^{2} \)
89 \( 1 + (0.962 + 0.269i)T^{2} \)
97 \( 1 + (-0.730 - 0.682i)T^{2} \)
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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

−8.661049250854631479921883988423, −8.076415902315967795271390711634, −6.77047335373285856757829485535, −6.54653072290675128403207946280, −5.60178813227597098451689025174, −5.05818209284217451665839465397, −4.33016712659773880536614112758, −3.72810619707336950821063430767, −2.94459501097401915480694703592, −1.93822253314729650232638386059, 1.46959888945428099886723419130, 2.24517075435508741981013699034, 3.09975979747717250651462649948, 4.07957144744387779997367921742, 4.77822920869441528518804432332, 5.78437889005106597587303482203, 6.01586728983713604106572517364, 6.82453179607959600063411965610, 7.55022854798545547423024629177, 8.095945314614839112112044075293

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