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Properties

Label 2-525-7.2-c1-0-22
Degree $2$
Conductor $525$
Sign $-0.905 + 0.424i$
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  + (1.21 − 2.11i)2-s + (0.5 + 0.866i)3-s + (−1.97 − 3.41i)4-s + 2.43·6-s + (−0.970 − 2.46i)7-s − 4.73·8-s + (−0.499 + 0.866i)9-s + (−2.29 − 3.96i)11-s + (1.97 − 3.41i)12-s − 1.35·13-s + (−6.37 − 0.951i)14-s + (−1.82 + 3.15i)16-s + (2.14 + 3.71i)17-s + (1.21 + 2.11i)18-s + (3.40 − 5.90i)19-s + ⋯
L(s)  = 1  + (0.861 − 1.49i)2-s + (0.288 + 0.499i)3-s + (−0.985 − 1.70i)4-s + 0.995·6-s + (−0.366 − 0.930i)7-s − 1.67·8-s + (−0.166 + 0.288i)9-s + (−0.690 − 1.19i)11-s + (0.568 − 0.985i)12-s − 0.376·13-s + (−1.70 − 0.254i)14-s + (−0.455 + 0.789i)16-s + (0.520 + 0.901i)17-s + (0.287 + 0.497i)18-s + (0.781 − 1.35i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.905 + 0.424i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.905 + 0.424i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.450098 - 2.01799i\)
\(L(\frac12)\) \(\approx\) \(0.450098 - 2.01799i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (0.970 + 2.46i)T \)
good2 \( 1 + (-1.21 + 2.11i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (2.29 + 3.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.35T + 13T^{2} \)
17 \( 1 + (-2.14 - 3.71i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.40 + 5.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.94 + 3.36i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (-4.05 - 7.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.87 - 10.1i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.17T + 41T^{2} \)
43 \( 1 + 3.52T + 43T^{2} \)
47 \( 1 + (1.64 - 2.85i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.72 - 8.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.350 + 0.606i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.970 - 1.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + (3.08 + 5.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.08 + 5.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + (0.146 - 0.253i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.00T + 97T^{2} \)
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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

−10.54784268740032476240930420749, −10.12290109004762822029547720628, −9.066849454631045979920990638198, −7.999629805171926102606633394193, −6.57170695511623306144788359043, −5.24106367014547324677679289101, −4.51894101028242584156238998031, −3.33318813516674120268379882559, −2.84027558550633241430122622834, −0.941099163757655074778257927121, 2.42224407478160468702517585864, 3.70941787362599407970629209820, 5.10598344838826113155359054127, 5.62138540309324910494320807257, 6.71913262842550218470626919277, 7.52840016218093237597794774884, 8.044742440027209668967497140983, 9.239428493477531297520624301758, 10.00466130567054952448844214904, 11.81189522561584724070709486080

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