↑

Properties

Label 2-525-35.27-c1-0-12
Degree $2$
Conductor $525$
Sign $-0.0153 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.68 + 1.68i)2-s + (−0.707 − 0.707i)3-s + 3.66i·4-s − 2.38i·6-s + (2.48 − 0.901i)7-s + (−2.81 + 2.81i)8-s + 1.00i·9-s + 1.54·11-s + (2.59 − 2.59i)12-s + (4.16 + 4.16i)13-s + (5.70 + 2.66i)14-s − 2.12·16-s + (−2.05 + 2.05i)17-s + (−1.68 + 1.68i)18-s − 5.70·19-s + ⋯
L(s)  = 1  + (1.19 + 1.19i)2-s + (−0.408 − 0.408i)3-s + 1.83i·4-s − 0.972i·6-s + (0.940 − 0.340i)7-s + (−0.993 + 0.993i)8-s + 0.333i·9-s + 0.465·11-s + (0.748 − 0.748i)12-s + (1.15 + 1.15i)13-s + (1.52 + 0.713i)14-s − 0.530·16-s + (−0.499 + 0.499i)17-s + (−0.396 + 0.396i)18-s − 1.30·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0153 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0153 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0153 - 0.999i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.0153 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81535 + 1.84347i\)
\(L(\frac12)\) \(\approx\) \(1.81535 + 1.84347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-2.48 + 0.901i)T \)
good2 \( 1 + (-1.68 - 1.68i)T + 2iT^{2} \)
11 \( 1 - 1.54T + 11T^{2} \)
13 \( 1 + (-4.16 - 4.16i)T + 13iT^{2} \)
17 \( 1 + (2.05 - 2.05i)T - 17iT^{2} \)
19 \( 1 + 5.70T + 19T^{2} \)
23 \( 1 + (3.72 - 3.72i)T - 23iT^{2} \)
29 \( 1 + 9.79iT - 29T^{2} \)
31 \( 1 + 3.81iT - 31T^{2} \)
37 \( 1 + (-2.81 - 2.81i)T + 37iT^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + (-1.02 + 1.02i)T - 43iT^{2} \)
47 \( 1 + (-2.05 + 2.05i)T - 47iT^{2} \)
53 \( 1 + (-4.89 + 4.89i)T - 53iT^{2} \)
59 \( 1 + 7.07T + 59T^{2} \)
61 \( 1 - 15.2iT - 61T^{2} \)
67 \( 1 + (5.31 + 5.31i)T + 67iT^{2} \)
71 \( 1 + 4.45T + 71T^{2} \)
73 \( 1 + (-0.770 - 0.770i)T + 73iT^{2} \)
79 \( 1 + 11.1iT - 79T^{2} \)
83 \( 1 + (9.60 + 9.60i)T + 83iT^{2} \)
89 \( 1 + 9.52T + 89T^{2} \)
97 \( 1 + (-5.09 + 5.09i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.51829543205914017941540071431, −10.44668796431100984586665670798, −8.839329254984869309615111400912, −8.069211338838061982803644975235, −7.19467620600083398741423580583, −6.27373722607814687016610276058, −5.80266340387261029729245626343, −4.27953914188193807373360904590, −4.12470535452176381113577569205, −1.86714545130457098078823223224, 1.36357843485821652086302750117, 2.74939209061236351796999988295, 3.93448850697904893307290843607, 4.72394584290751972405356440195, 5.59279383274081123446075804584, 6.46033657603253065228241101140, 8.164793922615463727513875031348, 9.045185130637075449443746903080, 10.39300484635132473259696448932, 10.86534776725497528757007678494

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