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Properties

Label 2-525-105.89-c1-0-14
Degree $2$
Conductor $525$
Sign $0.627 - 0.778i$
Analytic cond. $4.19214$
Root an. cond. $2.04747$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.442 + 0.766i)2-s + (−1.72 + 0.189i)3-s + (0.608 + 1.05i)4-s + (0.616 − 1.40i)6-s + (0.206 − 2.63i)7-s − 2.84·8-s + (2.92 − 0.652i)9-s + (1.25 − 0.723i)11-s + (−1.24 − 1.69i)12-s + 4.04·13-s + (1.93 + 1.32i)14-s + (0.0422 − 0.0730i)16-s + (4.98 − 2.87i)17-s + (−0.795 + 2.53i)18-s + (0.356 + 0.206i)19-s + ⋯
L(s)  = 1  + (−0.312 + 0.541i)2-s + (−0.993 + 0.109i)3-s + (0.304 + 0.527i)4-s + (0.251 − 0.572i)6-s + (0.0778 − 0.996i)7-s − 1.00·8-s + (0.976 − 0.217i)9-s + (0.377 − 0.218i)11-s + (−0.360 − 0.490i)12-s + 1.12·13-s + (0.515 + 0.354i)14-s + (0.0105 − 0.0182i)16-s + (1.20 − 0.698i)17-s + (−0.187 + 0.596i)18-s + (0.0818 + 0.0472i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.627 - 0.778i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.627 - 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.922336 + 0.441086i\)
\(L(\frac12)\) \(\approx\) \(0.922336 + 0.441086i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.189i)T \)
5 \( 1 \)
7 \( 1 + (-0.206 + 2.63i)T \)
good2 \( 1 + (0.442 - 0.766i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-1.25 + 0.723i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.04T + 13T^{2} \)
17 \( 1 + (-4.98 + 2.87i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.356 - 0.206i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.65 - 6.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.82iT - 29T^{2} \)
31 \( 1 + (-2.30 + 1.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.06 - 2.92i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.46T + 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + (-9.41 - 5.43i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.697 + 1.20i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.583 - 1.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.58 - 2.07i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.99 + 3.46i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.3iT - 71T^{2} \)
73 \( 1 + (1.57 + 2.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.42 - 11.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.5iT - 83T^{2} \)
89 \( 1 + (-3.90 + 6.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.86T + 97T^{2} \)
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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

−11.23930968197413419647584204734, −10.07103014956964550364525128202, −9.307799248612935726910789445012, −7.905768229595247554011731725903, −7.46184862991400723916076534348, −6.36138820606177766663146735333, −5.77447510782033837938197847562, −4.28949872700039730207447198644, −3.36561688871532411610237203826, −1.07371782584137017041327577074, 1.06183122767056694196046016569, 2.30072318264509836345543165752, 3.94605312940634658707804691654, 5.46800100112915420644089332088, 5.95825571425260714672087976417, 6.79838837481184432674191594414, 8.229049747822933419731043166278, 9.161958943124908325755297149254, 10.15170678098280271306345708687, 10.71607656596933627078032901732

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