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Properties

Label 2-525-105.2-c1-0-5
Degree $2$
Conductor $525$
Sign $0.619 - 0.784i$
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  + (−0.578 − 2.15i)2-s + (0.988 + 1.42i)3-s + (−2.59 + 1.5i)4-s + (2.5 − 2.95i)6-s + (−0.684 + 2.55i)7-s + (1.58 + 1.58i)8-s + (−1.04 + 2.81i)9-s + (−4.70 − 2.21i)12-s + (−3.74 + 3.74i)13-s + 5.91·14-s + (−0.500 + 0.866i)16-s + (−4.31 − 1.15i)17-s + (6.67 + 0.631i)18-s + (1.73 + i)19-s + (−4.31 + 1.55i)21-s + ⋯
L(s)  = 1  + (−0.409 − 1.52i)2-s + (0.570 + 0.821i)3-s + (−1.29 + 0.750i)4-s + (1.02 − 1.20i)6-s + (−0.258 + 0.965i)7-s + (0.559 + 0.559i)8-s + (−0.348 + 0.937i)9-s + (−1.35 − 0.638i)12-s + (−1.03 + 1.03i)13-s + 1.58·14-s + (−0.125 + 0.216i)16-s + (−1.04 − 0.280i)17-s + (1.57 + 0.148i)18-s + (0.397 + 0.229i)19-s + (−0.940 + 0.338i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.619 - 0.784i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.619 - 0.784i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.726584 + 0.352111i\)
\(L(\frac12)\) \(\approx\) \(0.726584 + 0.352111i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.988 - 1.42i)T \)
5 \( 1 \)
7 \( 1 + (0.684 - 2.55i)T \)
good2 \( 1 + (0.578 + 2.15i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.74 - 3.74i)T - 13iT^{2} \)
17 \( 1 + (4.31 + 1.15i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.73 - i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (6.47 - 1.73i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.91T + 29T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-10.2 + 2.73i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.91iT - 41T^{2} \)
43 \( 1 + (1.87 - 1.87i)T - 43iT^{2} \)
47 \( 1 + (-2.31 - 8.63i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.15 + 4.31i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.91 - 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.684 + 2.55i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + (-5.11 - 1.36i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.73 - i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.58 - 1.58i)T + 83iT^{2} \)
89 \( 1 + (2.95 - 5.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.48 + 7.48i)T + 97iT^{2} \)
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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

−11.03107791602564426270161035192, −9.768958431543170836423525374552, −9.642409027053956101027228562349, −8.820437656974294392622366915648, −7.901059477337125668850832698141, −6.30379283048531637390043351169, −4.83184837573553110057809531546, −3.99441957609587255020022755174, −2.72850403269101540113904453940, −2.12452011533495633744865113450, 0.47983542113944709752270530694, 2.60772980841184607160244342440, 4.19200303987719063392881875661, 5.50767250321716238357542860169, 6.57837538153544384107125857773, 7.10644498310521524581643052398, 7.932684804633820304945337811703, 8.482621205340455983947975818490, 9.586688258710071242851561218099, 10.31844428671761156623113142227

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