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Properties

Label 2-525-105.59-c1-0-5
Degree $2$
Conductor $525$
Sign $-0.628 + 0.777i$
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  + (1.16 + 2.01i)2-s + (−0.436 + 1.67i)3-s + (−1.71 + 2.97i)4-s + (−3.89 + 1.07i)6-s + (−2.39 + 1.11i)7-s − 3.33·8-s + (−2.61 − 1.46i)9-s + (−2.42 − 1.39i)11-s + (−4.23 − 4.17i)12-s + 3.20·13-s + (−5.04 − 3.53i)14-s + (−0.459 − 0.795i)16-s + (0.763 + 0.440i)17-s + (−0.0942 − 6.99i)18-s + (−1.90 + 1.09i)19-s + ⋯
L(s)  = 1  + (0.824 + 1.42i)2-s + (−0.252 + 0.967i)3-s + (−0.858 + 1.48i)4-s + (−1.58 + 0.437i)6-s + (−0.906 + 0.422i)7-s − 1.18·8-s + (−0.872 − 0.488i)9-s + (−0.729 − 0.421i)11-s + (−1.22 − 1.20i)12-s + 0.888·13-s + (−1.34 − 0.946i)14-s + (−0.114 − 0.198i)16-s + (0.185 + 0.106i)17-s + (−0.0222 − 1.64i)18-s + (−0.436 + 0.251i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.628 + 0.777i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (374, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.628 + 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.619873 - 1.29796i\)
\(L(\frac12)\) \(\approx\) \(0.619873 - 1.29796i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.436 - 1.67i)T \)
5 \( 1 \)
7 \( 1 + (2.39 - 1.11i)T \)
good2 \( 1 + (-1.16 - 2.01i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (2.42 + 1.39i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.20T + 13T^{2} \)
17 \( 1 + (-0.763 - 0.440i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.90 - 1.09i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.77 - 6.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 8.15iT - 29T^{2} \)
31 \( 1 + (7.62 + 4.40i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.352 + 0.203i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.55T + 41T^{2} \)
43 \( 1 - 0.118iT - 43T^{2} \)
47 \( 1 + (2.27 - 1.31i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.73 - 6.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.04 + 3.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.7 + 6.17i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.38 - 0.802i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.25iT - 71T^{2} \)
73 \( 1 + (-0.110 + 0.192i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.56 + 2.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.666iT - 83T^{2} \)
89 \( 1 + (0.437 + 0.757i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.37T + 97T^{2} \)
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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

−11.34025675783133075146542476117, −10.58077451776875402949478866749, −9.378602432318721841144932541517, −8.714289788660681838652740799703, −7.66875441753777855688527848953, −6.54858613834635312905121385188, −5.71642240327454085529839549243, −5.26023049590410236936912818191, −3.91557899819704853974135656197, −3.21446422261850286771412290914, 0.66435121182094783907499182189, 2.17787369666070758452065686976, 3.10741177954429483438830461933, 4.28783864257816736814575414261, 5.45794541879557313981651365659, 6.42873893922622829146760937036, 7.42016378359728537541340019347, 8.648603670472417199451105906834, 9.813977944979682373681516993517, 10.70208916752805691164482585979

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