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Properties

Label 2-525-105.23-c1-0-3
Degree $2$
Conductor $525$
Sign $0.370 - 0.928i$
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  + (0.298 − 0.0799i)2-s + (−1.15 − 1.29i)3-s + (−1.64 + 0.952i)4-s + (−0.447 − 0.292i)6-s + (−0.951 − 2.46i)7-s + (−0.852 + 0.852i)8-s + (−0.333 + 2.98i)9-s + (−0.660 + 0.381i)11-s + (3.13 + 1.02i)12-s + (2.27 + 2.27i)13-s + (−0.481 − 0.660i)14-s + (1.71 − 2.97i)16-s + (−1.25 + 4.69i)17-s + (0.138 + 0.916i)18-s + (1.41 + 0.818i)19-s + ⋯
L(s)  = 1  + (0.210 − 0.0565i)2-s + (−0.666 − 0.745i)3-s + (−0.824 + 0.476i)4-s + (−0.182 − 0.119i)6-s + (−0.359 − 0.933i)7-s + (−0.301 + 0.301i)8-s + (−0.111 + 0.993i)9-s + (−0.199 + 0.114i)11-s + (0.904 + 0.297i)12-s + (0.629 + 0.629i)13-s + (−0.128 − 0.176i)14-s + (0.429 − 0.744i)16-s + (−0.305 + 1.13i)17-s + (0.0327 + 0.215i)18-s + (0.325 + 0.187i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.370 - 0.928i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.370 - 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555957 + 0.376724i\)
\(L(\frac12)\) \(\approx\) \(0.555957 + 0.376724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.15 + 1.29i)T \)
5 \( 1 \)
7 \( 1 + (0.951 + 2.46i)T \)
good2 \( 1 + (-0.298 + 0.0799i)T + (1.73 - i)T^{2} \)
11 \( 1 + (0.660 - 0.381i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.27 - 2.27i)T + 13iT^{2} \)
17 \( 1 + (1.25 - 4.69i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.41 - 0.818i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.98 - 7.39i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 4.94T + 29T^{2} \)
31 \( 1 + (-2.96 - 5.13i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.915 - 3.41i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.35iT - 41T^{2} \)
43 \( 1 + (2.69 + 2.69i)T + 43iT^{2} \)
47 \( 1 + (-4.14 + 1.10i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.71 + 1.79i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.84 - 6.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.19 - 3.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0471 - 0.0126i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 + (0.359 - 1.34i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.66 + 2.11i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.05 + 5.05i)T - 83iT^{2} \)
89 \( 1 + (0.453 - 0.785i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.73 - 3.73i)T - 97iT^{2} \)
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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

−11.15129502544957468835941195551, −10.27110397799190399965876378117, −9.229741896602550937235692593753, −8.176680675060025673978867049912, −7.41403046987232389759557179424, −6.46861383668340049661122056412, −5.43594922050468980050562361175, −4.34632515126166257961935233866, −3.36319522253013693040732836232, −1.40931617314454220643852878255, 0.44471502070475327960825321061, 2.91167385050715183840474195931, 4.16233910395496000897555250747, 5.16016659569219196091240431606, 5.76196282402770943054593086255, 6.66198099582122016398546634727, 8.317654474580209023412097674031, 9.181758716264781336951344218411, 9.694880646651712388481244891659, 10.68420653507069971388764258115

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