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Properties

Label 2-525-21.5-c1-0-7
Degree $2$
Conductor $525$
Sign $-0.601 - 0.799i$
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  + (0.558 + 0.322i)2-s + (0.358 + 1.69i)3-s + (−0.792 − 1.37i)4-s + (−0.346 + 1.06i)6-s + (0.105 + 2.64i)7-s − 2.31i·8-s + (−2.74 + 1.21i)9-s + (−3.51 + 2.02i)11-s + (2.04 − 1.83i)12-s + 4.21i·13-s + (−0.793 + 1.51i)14-s + (−0.839 + 1.45i)16-s + (1.08 + 1.88i)17-s + (−1.92 − 0.206i)18-s + (3.87 + 2.23i)19-s + ⋯
L(s)  = 1  + (0.394 + 0.227i)2-s + (0.206 + 0.978i)3-s + (−0.396 − 0.685i)4-s + (−0.141 + 0.433i)6-s + (0.0397 + 0.999i)7-s − 0.817i·8-s + (−0.914 + 0.404i)9-s + (−1.05 + 0.611i)11-s + (0.589 − 0.529i)12-s + 1.16i·13-s + (−0.212 + 0.403i)14-s + (−0.209 + 0.363i)16-s + (0.263 + 0.457i)17-s + (−0.453 − 0.0486i)18-s + (0.889 + 0.513i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.601 - 0.799i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.601 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.594080 + 1.19036i\)
\(L(\frac12)\) \(\approx\) \(0.594080 + 1.19036i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.358 - 1.69i)T \)
5 \( 1 \)
7 \( 1 + (-0.105 - 2.64i)T \)
good2 \( 1 + (-0.558 - 0.322i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (3.51 - 2.02i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.21iT - 13T^{2} \)
17 \( 1 + (-1.08 - 1.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.87 - 2.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.558 - 0.322i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.16iT - 29T^{2} \)
31 \( 1 + (0.339 - 0.195i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.13 + 3.69i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.27T + 41T^{2} \)
43 \( 1 - 6.54T + 43T^{2} \)
47 \( 1 + (-3.90 + 6.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.23 - 3.60i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.66 + 9.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.05 - 3.49i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.36 - 7.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.13iT - 71T^{2} \)
73 \( 1 + (-4.53 + 2.61i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.87 + 3.24i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.27T + 83T^{2} \)
89 \( 1 + (0.447 - 0.774i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.89iT - 97T^{2} \)
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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

−11.00304579798599976752866809165, −10.12851789231148087251781930036, −9.454630691344594447719837538564, −8.803888379263841740044134106616, −7.64475131047191915646000885855, −6.22284379333349348081826615057, −5.36881456617457625281879071923, −4.74958825446746813413878665825, −3.61796495584761916262047883969, −2.15110976703510477005609873866, 0.67341511464437379827513760358, 2.72871051105928356944465149105, 3.38693178197801946779027009133, 4.85682045503291331388911963399, 5.80191090163435882132357396882, 7.24226285722209024384944049962, 7.78583885213026862335487836895, 8.423433561888279731058692565991, 9.636084052719995213926214424091, 10.83626407050261683111412105605

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