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Properties

Label 2-525-35.24-c2-0-39
Degree $2$
Conductor $525$
Sign $-0.742 + 0.669i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.25 + 0.724i)2-s + (0.866 − 1.5i)3-s + (−0.949 + 1.64i)4-s + 2.51i·6-s + (2.59 − 6.5i)7-s − 8.55i·8-s + (−1.5 − 2.59i)9-s + (3.44 − 5.97i)11-s + (1.64 + 2.84i)12-s − 0.174·13-s + (1.44 + 10.0i)14-s + (2.39 + 4.15i)16-s + (−9.43 + 16.3i)17-s + (3.76 + 2.17i)18-s + (−29.5 + 17.0i)19-s + ⋯
L(s)  = 1  + (−0.627 + 0.362i)2-s + (0.288 − 0.5i)3-s + (−0.237 + 0.411i)4-s + 0.418i·6-s + (0.371 − 0.928i)7-s − 1.06i·8-s + (−0.166 − 0.288i)9-s + (0.313 − 0.543i)11-s + (0.137 + 0.237i)12-s − 0.0134·13-s + (0.103 + 0.717i)14-s + (0.149 + 0.259i)16-s + (−0.555 + 0.961i)17-s + (0.209 + 0.120i)18-s + (−1.55 + 0.897i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.742 + 0.669i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.742 + 0.669i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4799228459\)
\(L(\frac12)\) \(\approx\) \(0.4799228459\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 \)
7 \( 1 + (-2.59 + 6.5i)T \)
good2 \( 1 + (1.25 - 0.724i)T + (2 - 3.46i)T^{2} \)
11 \( 1 + (-3.44 + 5.97i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 0.174T + 169T^{2} \)
17 \( 1 + (9.43 - 16.3i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (29.5 - 17.0i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-17.3 + 10i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 27.3T + 841T^{2} \)
31 \( 1 + (29.3 + 16.9i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (0.437 - 0.252i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 42.0iT - 1.68e3T^{2} \)
43 \( 1 + 40.2iT - 1.84e3T^{2} \)
47 \( 1 + (8.05 + 13.9i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (44.5 + 25.7i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-72.1 - 41.6i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (80.5 - 46.5i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (79.7 + 46.0i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 74.9T + 5.04e3T^{2} \)
73 \( 1 + (4.50 - 7.80i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (29.0 + 50.3i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 83.9T + 6.88e3T^{2} \)
89 \( 1 + (-54.7 + 31.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 131.T + 9.40e3T^{2} \)
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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

−10.32765175683251307185799959403, −9.027681908293839023096935363893, −8.526736255168628225940000791921, −7.68375250920893409112134694763, −6.94030900406495729114576830945, −6.01609926353853839025445547228, −4.28181959458922744997573212837, −3.58349353891571539866007611520, −1.76443782589107340967119934739, −0.22072295295459455326788367443, 1.75026718698313211880926917767, 2.77626337374255937124865155988, 4.52145881720191889330983652256, 5.15549357120744473973751812851, 6.36494510525285689375200600346, 7.67008919170051087786473715379, 8.888125174922843769840225628889, 9.064806186970860808758185744240, 9.923898310448844639824151067492, 11.11577063419691629746928813320

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