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Properties

Label 2-525-7.4-c1-0-15
Degree $2$
Conductor $525$
Sign $0.910 - 0.413i$
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  + (0.614 + 1.06i)2-s + (−0.5 + 0.866i)3-s + (0.245 − 0.424i)4-s − 1.22·6-s + (−1.24 − 2.33i)7-s + 3.05·8-s + (−0.499 − 0.866i)9-s + (2.37 − 4.11i)11-s + (0.245 + 0.424i)12-s + 6.25·13-s + (1.71 − 2.75i)14-s + (1.38 + 2.40i)16-s + (−3.14 + 5.44i)17-s + (0.614 − 1.06i)18-s + (−2.47 − 4.28i)19-s + ⋯
L(s)  = 1  + (0.434 + 0.752i)2-s + (−0.288 + 0.499i)3-s + (0.122 − 0.212i)4-s − 0.501·6-s + (−0.470 − 0.882i)7-s + 1.08·8-s + (−0.166 − 0.288i)9-s + (0.715 − 1.23i)11-s + (0.0707 + 0.122i)12-s + 1.73·13-s + (0.459 − 0.737i)14-s + (0.347 + 0.601i)16-s + (−0.762 + 1.32i)17-s + (0.144 − 0.250i)18-s + (−0.567 − 0.983i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 - 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.910 - 0.413i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.910 - 0.413i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82493 + 0.395301i\)
\(L(\frac12)\) \(\approx\) \(1.82493 + 0.395301i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (1.24 + 2.33i)T \)
good2 \( 1 + (-0.614 - 1.06i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (-2.37 + 4.11i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.25T + 13T^{2} \)
17 \( 1 + (3.14 - 5.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.47 + 4.28i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.49 - 4.31i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (0.829 - 1.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.21 + 3.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.30T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + (-1.88 - 3.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.60 + 6.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.117 - 0.203i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.24 + 2.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.06T + 71T^{2} \)
73 \( 1 + (6.24 - 10.8i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.415 - 0.719i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.14T + 83T^{2} \)
89 \( 1 + (1.14 + 1.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.476T + 97T^{2} \)
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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

−10.96066851763032168403542637499, −10.29186941305697626641846096619, −8.993966952031965119746090153673, −8.279858350933432084231758260485, −6.76812105618882955660255535793, −6.40331923048061145521389837567, −5.52767163729549852000302800888, −4.21085731265337432319230385533, −3.55378076460218338060247354729, −1.19207250637030314318361003498, 1.61556213676228807284262139541, 2.69200180244967115943149466770, 3.90544396125159232252391776142, 4.96887990097747068062917677051, 6.37006798814638598663227857713, 6.89788110961301325978261960785, 8.202475060167686273509522098050, 9.021494252079550494848361052043, 10.16133366317719413051268694478, 11.07137265124170214533896358401

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