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Properties

Label 2-525-35.4-c1-0-6
Degree $2$
Conductor $525$
Sign $0.629 - 0.777i$
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  + (1.06 − 0.614i)2-s + (−0.866 − 0.5i)3-s + (−0.245 + 0.424i)4-s − 1.22·6-s + (−2.33 + 1.24i)7-s + 3.05i·8-s + (0.499 + 0.866i)9-s + (2.37 − 4.11i)11-s + (0.424 − 0.245i)12-s + 6.25i·13-s + (−1.71 + 2.75i)14-s + (1.38 + 2.40i)16-s + (5.44 + 3.14i)17-s + (1.06 + 0.614i)18-s + (2.47 + 4.28i)19-s + ⋯
L(s)  = 1  + (0.752 − 0.434i)2-s + (−0.499 − 0.288i)3-s + (−0.122 + 0.212i)4-s − 0.501·6-s + (−0.882 + 0.470i)7-s + 1.08i·8-s + (0.166 + 0.288i)9-s + (0.715 − 1.23i)11-s + (0.122 − 0.0707i)12-s + 1.73i·13-s + (−0.459 + 0.737i)14-s + (0.347 + 0.601i)16-s + (1.32 + 0.762i)17-s + (0.250 + 0.144i)18-s + (0.567 + 0.983i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.629 - 0.777i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.629 - 0.777i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28209 + 0.611656i\)
\(L(\frac12)\) \(\approx\) \(1.28209 + 0.611656i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−11.41503623442915824733503571131, −10.25838623015889223420872813714, −9.192712715249472211277464317928, −8.411698047735482938419898377519, −7.21688549500468865342844145941, −6.00707798069802166041056014307, −5.57925501501405448371529803332, −3.96344443098882675472664037534, −3.43048502762044742419254912501, −1.79636395803195273539293131317, 0.73598118507422499887183694827, 3.17526621901850967626510559408, 4.18030162958167442631543550832, 5.18922198796111721292229937768, 5.91366534150149590622774862125, 6.92758897963372040775562554618, 7.61528606407205068967313880826, 9.379043777585969482231946916392, 9.880930781562086004805826401807, 10.49693682477330954307239978454

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