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Properties

Label 2-525-21.20-c1-0-26
Degree $2$
Conductor $525$
Sign $0.537 + 0.843i$
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  + 1.73i·2-s + (−1.41 − i)3-s − 0.999·4-s + (1.73 − 2.44i)6-s + (−2.44 − i)7-s + 1.73i·8-s + (1.00 + 2.82i)9-s − 2.82i·11-s + (1.41 + 0.999i)12-s − 4i·13-s + (1.73 − 4.24i)14-s − 5·16-s − 2.82·17-s + (−4.89 + 1.73i)18-s + (2.46 + 3.86i)21-s + 4.89·22-s + ⋯
L(s)  = 1  + 1.22i·2-s + (−0.816 − 0.577i)3-s − 0.499·4-s + (0.707 − 0.999i)6-s + (−0.925 − 0.377i)7-s + 0.612i·8-s + (0.333 + 0.942i)9-s − 0.852i·11-s + (0.408 + 0.288i)12-s − 1.10i·13-s + (0.462 − 1.13i)14-s − 1.25·16-s − 0.685·17-s + (−1.15 + 0.408i)18-s + (0.537 + 0.843i)21-s + 1.04·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.537 + 0.843i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ 0.537 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.507870 - 0.278466i\)
\(L(\frac12)\) \(\approx\) \(0.507870 - 0.278466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.41 + i)T \)
5 \( 1 \)
7 \( 1 + (2.44 + i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 + 9.79iT - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 - 4.89T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 - 9.79iT - 61T^{2} \)
67 \( 1 - 4.89T + 67T^{2} \)
71 \( 1 + 2.82iT - 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 2.82T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−10.82537480129378212773993164572, −9.842119214266647048228940807495, −8.527026226510896588755644854835, −7.76609646723038776318040430757, −6.94254914315368758352926512651, −6.08982515924491753361238961010, −5.69112696278404865574038099762, −4.34921749087422178565706792364, −2.62678131289774641261413808408, −0.36422796138032839689532786261, 1.67990065003813882294903981707, 3.13270138117997919483161469832, 4.10966826692915415762069196150, 5.10368955958346986016168066349, 6.52640397618182810756026004890, 6.97353604084205840388794881896, 9.027436397531552309515880943751, 9.466144426524790327033491114834, 10.29402773762022808284365950999, 10.98810205377430434646893258859

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