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Properties

Label 2-525-5.3-c2-0-16
Degree $2$
Conductor $525$
Sign $0.850 - 0.525i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.675 + 0.675i)2-s + (−1.22 − 1.22i)3-s + 3.08i·4-s + 1.65·6-s + (1.87 − 1.87i)7-s + (−4.78 − 4.78i)8-s + 2.99i·9-s + 7.59·11-s + (3.78 − 3.78i)12-s + (−1.12 − 1.12i)13-s + 2.52i·14-s − 5.88·16-s + (3.43 − 3.43i)17-s + (−2.02 − 2.02i)18-s − 26.3i·19-s + ⋯
L(s)  = 1  + (−0.337 + 0.337i)2-s + (−0.408 − 0.408i)3-s + 0.771i·4-s + 0.275·6-s + (0.267 − 0.267i)7-s + (−0.598 − 0.598i)8-s + 0.333i·9-s + 0.690·11-s + (0.315 − 0.315i)12-s + (−0.0863 − 0.0863i)13-s + 0.180i·14-s − 0.367·16-s + (0.202 − 0.202i)17-s + (−0.112 − 0.112i)18-s − 1.38i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.281142543\)
\(L(\frac12)\) \(\approx\) \(1.281142543\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.22 + 1.22i)T \)
5 \( 1 \)
7 \( 1 + (-1.87 + 1.87i)T \)
good2 \( 1 + (0.675 - 0.675i)T - 4iT^{2} \)
11 \( 1 - 7.59T + 121T^{2} \)
13 \( 1 + (1.12 + 1.12i)T + 169iT^{2} \)
17 \( 1 + (-3.43 + 3.43i)T - 289iT^{2} \)
19 \( 1 + 26.3iT - 361T^{2} \)
23 \( 1 + (-24.2 - 24.2i)T + 529iT^{2} \)
29 \( 1 - 22.3iT - 841T^{2} \)
31 \( 1 + 18.3T + 961T^{2} \)
37 \( 1 + (-34.4 + 34.4i)T - 1.36e3iT^{2} \)
41 \( 1 - 37.4T + 1.68e3T^{2} \)
43 \( 1 + (-55.1 - 55.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (40.8 - 40.8i)T - 2.20e3iT^{2} \)
53 \( 1 + (-9.39 - 9.39i)T + 2.80e3iT^{2} \)
59 \( 1 - 49.7iT - 3.48e3T^{2} \)
61 \( 1 - 88.3T + 3.72e3T^{2} \)
67 \( 1 + (-40.5 + 40.5i)T - 4.48e3iT^{2} \)
71 \( 1 - 136.T + 5.04e3T^{2} \)
73 \( 1 + (5.85 + 5.85i)T + 5.32e3iT^{2} \)
79 \( 1 - 66.4iT - 6.24e3T^{2} \)
83 \( 1 + (-34.8 - 34.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 2.03iT - 7.92e3T^{2} \)
97 \( 1 + (-58.4 + 58.4i)T - 9.40e3iT^{2} \)
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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

−11.16787621084735851517873979060, −9.504192527249823862665445418909, −8.997983346481478524954608867375, −7.79970149554581560518943345197, −7.22198325149181472547775854961, −6.42904331724151544204569769322, −5.16240986275290408058106194931, −3.99124770810491365675730336813, −2.72739650176075865072511958890, −0.923866543953594248807213345447, 0.870744730585352357570085083187, 2.22735331234347134400048381293, 3.84440873761273847526024297663, 5.00372727843991223464791080407, 5.88046405518761274818485682975, 6.69143323560569214209344422186, 8.134671423698044356376559586035, 9.049512893318799876899572097454, 9.799740061399449149431720383042, 10.53279977470007738552819557968

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