Properties

Label 2-525-105.2-c1-0-23
Degree $2$
Conductor $525$
Sign $-0.442 - 0.896i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 1.36i)2-s + (0.599 + 1.62i)3-s + (−2 + 1.41i)6-s + (2.38 + 1.15i)7-s + (2 + 1.99i)8-s + (−2.28 + 1.94i)9-s + (3.67 − 2.12i)11-s + (3.53 − 3.53i)13-s + (−0.707 + 3.67i)14-s + (−1.99 + 3.46i)16-s + (−1.36 − 0.366i)17-s + (−3.49 − 2.40i)18-s + (−6.06 − 3.5i)19-s + (−0.449 + 4.56i)21-s + (4.24 + 4.24i)22-s + (−4.09 + 1.09i)23-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (0.346 + 0.938i)3-s + (−0.816 + 0.577i)6-s + (0.899 + 0.436i)7-s + (0.707 + 0.707i)8-s + (−0.760 + 0.649i)9-s + (1.10 − 0.639i)11-s + (0.980 − 0.980i)13-s + (−0.188 + 0.981i)14-s + (−0.499 + 0.866i)16-s + (−0.331 − 0.0887i)17-s + (−0.824 − 0.565i)18-s + (−1.39 − 0.802i)19-s + (−0.0980 + 0.995i)21-s + (0.904 + 0.904i)22-s + (−0.854 + 0.228i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.442 - 0.896i$
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: \((2,\ 525,\ (\ :1/2),\ -0.442 - 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19934 + 1.93000i\)
\(L(\frac12)\) \(\approx\) \(1.19934 + 1.93000i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.599 - 1.62i)T \)
5 \( 1 \)
7 \( 1 + (-2.38 - 1.15i)T \)
good2 \( 1 + (-0.366 - 1.36i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-3.67 + 2.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.53 + 3.53i)T - 13iT^{2} \)
17 \( 1 + (1.36 + 0.366i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.06 + 3.5i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.09 - 1.09i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.07T + 29T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.965 - 0.258i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 7.07iT - 41T^{2} \)
43 \( 1 + (3.53 - 3.53i)T - 43iT^{2} \)
47 \( 1 + (1.46 + 5.46i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.732 - 2.73i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.258 - 0.965i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 7.07iT - 71T^{2} \)
73 \( 1 + (-6.76 - 1.81i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.06 + 3.5i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (1 + i)T + 83iT^{2} \)
89 \( 1 + (-7.77 + 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.07 - 7.07i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.07275974883572719925078860601, −10.41804942393756231246473036111, −8.937611171956673177506876007098, −8.552312275063666243138076169425, −7.65950477524853648005267674862, −6.30823519245385903726049541796, −5.65289180309243924654381663362, −4.67337617088692544196036024277, −3.66321057162493646453210492360, −2.03643403302992216206165889242, 1.54599430225155699533172763868, 1.99832383011435197432469926491, 3.74210156812899871120782348219, 4.31655917767988480843085532245, 6.22642503457746243542700639708, 6.90564559929241860474913024856, 7.903924712636153431284273344175, 8.748775814141438269033775357056, 9.813165728062686435509191962300, 11.01056574153344517019221955383

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