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Properties

Label 2-525-105.32-c1-0-21
Degree $2$
Conductor $525$
Sign $0.999 - 0.00489i$
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  + (1.36 + 0.366i)2-s + (−1.62 − 0.599i)3-s + (−2 − 1.41i)6-s + (1.15 + 2.38i)7-s + (−1.99 − 2i)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 + (−0.366 − 1.36i)17-s + (2.40 + 3.49i)18-s + (6.06 − 3.5i)19-s + (−0.449 − 4.56i)21-s + (4.24 + 4.24i)22-s + (−1.09 + 4.09i)23-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.938 − 0.346i)3-s + (−0.816 − 0.577i)6-s + (0.436 + 0.899i)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 + (−0.0887 − 0.331i)17-s + (0.565 + 0.824i)18-s + (1.39 − 0.802i)19-s + (−0.0980 − 0.995i)21-s + (0.904 + 0.904i)22-s + (−0.228 + 0.854i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.999 - 0.00489i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.999 - 0.00489i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79435 + 0.00439070i\)
\(L(\frac12)\) \(\approx\) \(1.79435 + 0.00439070i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62 + 0.599i)T \)
5 \( 1 \)
7 \( 1 + (-1.15 - 2.38i)T \)
good2 \( 1 + (-1.36 - 0.366i)T + (1.73 + i)T^{2} \)
11 \( 1 + (-3.67 - 2.12i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.53 + 3.53i)T - 13iT^{2} \)
17 \( 1 + (0.366 + 1.36i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-6.06 + 3.5i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.09 - 4.09i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.07T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.258 + 0.965i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + (3.53 - 3.53i)T - 43iT^{2} \)
47 \( 1 + (5.46 + 1.46i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.73 - 0.732i)T + (45.8 - 26.5i)T^{2} \)
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 \( 1 + (-0.965 + 0.258i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.07iT - 71T^{2} \)
73 \( 1 + (1.81 + 6.76i)T + (-63.2 + 36.5i)T^{2} \)
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 \( 1 + (-7.07 - 7.07i)T + 97iT^{2} \)
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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

−11.35818179862862818045109003756, −9.966381364545234700558244019145, −9.212205593717541308882871969530, −7.981786557703790366262590796891, −6.79744360938483769107556379323, −6.06069162554419381493147543001, −5.25181065013024847749264872338, −4.55897040463544383027393430408, −3.17097766642364988303708366285, −1.23198412888851821063801694677, 1.26898138976229541699222355016, 3.57807404747232886283128585675, 4.09166026862852091018255908299, 5.02193726220212601735351354675, 6.10485727744469364245808440700, 6.76831668368101943834986527129, 8.249510185789761034023272552090, 9.168257573899771103024026478021, 10.28409479299802049929048265053, 11.15902021125722847481586474270

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