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Properties

Label 2-525-35.12-c1-0-16
Degree $2$
Conductor $525$
Sign $0.975 + 0.218i$
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  + (2.24 − 0.602i)2-s + (−0.258 + 0.965i)3-s + (2.95 − 1.70i)4-s + 2.32i·6-s + (2.59 − 0.519i)7-s + (2.33 − 2.33i)8-s + (−0.866 − 0.499i)9-s + (−1.76 − 3.05i)11-s + (0.884 + 3.30i)12-s + (4.49 + 4.49i)13-s + (5.51 − 2.73i)14-s + (0.421 − 0.729i)16-s + (−1.79 − 0.481i)17-s + (−2.24 − 0.602i)18-s + (0.0699 − 0.121i)19-s + ⋯
L(s)  = 1  + (1.58 − 0.425i)2-s + (−0.149 + 0.557i)3-s + (1.47 − 0.854i)4-s + 0.950i·6-s + (0.980 − 0.196i)7-s + (0.824 − 0.824i)8-s + (−0.288 − 0.166i)9-s + (−0.531 − 0.921i)11-s + (0.255 + 0.952i)12-s + (1.24 + 1.24i)13-s + (1.47 − 0.730i)14-s + (0.105 − 0.182i)16-s + (−0.436 − 0.116i)17-s + (−0.529 − 0.141i)18-s + (0.0160 − 0.0277i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.975 + 0.218i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.975 + 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.40880 - 0.377779i\)
\(L(\frac12)\) \(\approx\) \(3.40880 - 0.377779i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-2.59 + 0.519i)T \)
good2 \( 1 + (-2.24 + 0.602i)T + (1.73 - i)T^{2} \)
11 \( 1 + (1.76 + 3.05i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.49 - 4.49i)T + 13iT^{2} \)
17 \( 1 + (1.79 + 0.481i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.0699 + 0.121i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.997 + 3.72i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.01iT - 29T^{2} \)
31 \( 1 + (4.56 - 2.63i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.61 - 1.50i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.903iT - 41T^{2} \)
43 \( 1 + (2.38 - 2.38i)T - 43iT^{2} \)
47 \( 1 + (-0.639 - 2.38i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.71 + 0.726i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.15 + 5.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.69 + 5.01i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.77 + 10.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 5.09T + 71T^{2} \)
73 \( 1 + (2.42 - 9.04i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.30 + 4.21i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.37 - 7.37i)T + 83iT^{2} \)
89 \( 1 + (1.75 - 3.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.70 - 8.70i)T - 97iT^{2} \)
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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

−11.08328002112905626581339710456, −10.57290997924961370341122802747, −9.033134473640886024737209888230, −8.235919622481819492175685198233, −6.70632211548233231062657514482, −5.87085318312192755932377030661, −4.92154191331864789142425126256, −4.19409492574861971114613432464, −3.26994842435677556509027046141, −1.83275414716855389790175032761, 1.85173727240845257915398700529, 3.20885839200609436081409404294, 4.39914133016634969280564832328, 5.39606894111916872260936684728, 5.91081138691055004025754165857, 7.15537231834026244240873509759, 7.77304093422078551622648403893, 8.796311775538684485832713887614, 10.41834053222081579068049638031, 11.23237050781915618065343197717

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