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Properties

Label 2-525-35.33-c1-0-18
Degree $2$
Conductor $525$
Sign $0.441 + 0.897i$
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  + (−0.105 − 0.394i)2-s + (0.965 + 0.258i)3-s + (1.58 − 0.916i)4-s − 0.408i·6-s + (−2.57 − 0.605i)7-s + (−1.10 − 1.10i)8-s + (0.866 + 0.499i)9-s + (−0.463 − 0.803i)11-s + (1.77 − 0.474i)12-s + (4.08 − 4.08i)13-s + (0.0332 + 1.08i)14-s + (1.51 − 2.62i)16-s + (0.192 − 0.719i)17-s + (0.105 − 0.394i)18-s + (1.21 − 2.11i)19-s + ⋯
L(s)  = 1  + (−0.0747 − 0.278i)2-s + (0.557 + 0.149i)3-s + (0.793 − 0.458i)4-s − 0.166i·6-s + (−0.973 − 0.228i)7-s + (−0.391 − 0.391i)8-s + (0.288 + 0.166i)9-s + (−0.139 − 0.242i)11-s + (0.511 − 0.136i)12-s + (1.13 − 1.13i)13-s + (0.00889 + 0.288i)14-s + (0.378 − 0.655i)16-s + (0.0467 − 0.174i)17-s + (0.0249 − 0.0929i)18-s + (0.279 − 0.484i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.441 + 0.897i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.441 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56247 - 0.972166i\)
\(L(\frac12)\) \(\approx\) \(1.56247 - 0.972166i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (2.57 + 0.605i)T \)
good2 \( 1 + (0.105 + 0.394i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (0.463 + 0.803i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.08 + 4.08i)T - 13iT^{2} \)
17 \( 1 + (-0.192 + 0.719i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.21 + 2.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.00 + 1.34i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 8.08iT - 29T^{2} \)
31 \( 1 + (1.05 - 0.607i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.472 - 1.76i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 6.97iT - 41T^{2} \)
43 \( 1 + (-0.781 - 0.781i)T + 43iT^{2} \)
47 \( 1 + (10.0 - 2.70i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.72 - 6.42i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.91 + 10.2i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.72 + 2.15i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.4 - 2.80i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 + (4.02 + 1.07i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (7.02 + 4.05i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.91 - 5.91i)T - 83iT^{2} \)
89 \( 1 + (7.78 - 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.89 - 4.89i)T + 97iT^{2} \)
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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

−10.75830026674331646433900051178, −9.875954556502251332225377126397, −9.116027840236282773632074017866, −8.060803482059470977928639915478, −6.97501314584360784061902862757, −6.23588892590469859217209574629, −5.11993145054792464928327820337, −3.36843209817648434593990974304, −2.92654357934767694717294767891, −1.12982273537291892939332138098, 1.89912858962416989162734571039, 3.11273869244145091064215599576, 3.99789040299615218373202952743, 5.79419930856181416961845892870, 6.59530203471287086365354462603, 7.31459954651442800154088544210, 8.345129619220698700741811195177, 9.094861328306913683110470650838, 9.991932508762966450068267215827, 11.13821991767497258230442954576

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