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Properties

Label 2-525-15.2-c1-0-9
Degree $2$
Conductor $525$
Sign $-0.259 - 0.965i$
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  + (0.560 − 0.560i)2-s + (0.0537 + 1.73i)3-s + 1.37i·4-s + (0.999 + 0.939i)6-s + (−0.707 − 0.707i)7-s + (1.88 + 1.88i)8-s + (−2.99 + 0.186i)9-s + 4.10i·11-s + (−2.37 + 0.0737i)12-s + (1.67 − 1.67i)13-s − 0.792·14-s − 0.627·16-s + (−0.664 + 0.664i)17-s + (−1.57 + 1.78i)18-s + 4i·19-s + ⋯
L(s)  = 1  + (0.396 − 0.396i)2-s + (0.0310 + 0.999i)3-s + 0.686i·4-s + (0.408 + 0.383i)6-s + (−0.267 − 0.267i)7-s + (0.667 + 0.667i)8-s + (−0.998 + 0.0620i)9-s + 1.23i·11-s + (−0.685 + 0.0212i)12-s + (0.465 − 0.465i)13-s − 0.211·14-s − 0.156·16-s + (−0.161 + 0.161i)17-s + (−0.370 + 0.419i)18-s + 0.917i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.259 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.259 - 0.965i$
Analytic conductor: \(4.19214\)
Root analytic conductor: \(2.04747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1/2),\ -0.259 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.952452 + 1.24263i\)
\(L(\frac12)\) \(\approx\) \(0.952452 + 1.24263i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0537 - 1.73i)T \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-0.560 + 0.560i)T - 2iT^{2} \)
11 \( 1 - 4.10iT - 11T^{2} \)
13 \( 1 + (-1.67 + 1.67i)T - 13iT^{2} \)
17 \( 1 + (0.664 - 0.664i)T - 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (2.44 + 2.44i)T + 23iT^{2} \)
29 \( 1 + 5.98T + 29T^{2} \)
31 \( 1 - 4.74T + 31T^{2} \)
37 \( 1 + (-3.35 - 3.35i)T + 37iT^{2} \)
41 \( 1 - 11.9iT - 41T^{2} \)
43 \( 1 + (-5.65 + 5.65i)T - 43iT^{2} \)
47 \( 1 + (-3.11 + 3.11i)T - 47iT^{2} \)
53 \( 1 + (8.46 + 8.46i)T + 53iT^{2} \)
59 \( 1 - 11.9T + 59T^{2} \)
61 \( 1 - 6.74T + 61T^{2} \)
67 \( 1 + (-9.01 - 9.01i)T + 67iT^{2} \)
71 \( 1 + 1.87iT - 71T^{2} \)
73 \( 1 + (-9.01 + 9.01i)T - 73iT^{2} \)
79 \( 1 + 15.1iT - 79T^{2} \)
83 \( 1 + (-11.8 - 11.8i)T + 83iT^{2} \)
89 \( 1 - 3.75T + 89T^{2} \)
97 \( 1 + (1.67 + 1.67i)T + 97iT^{2} \)
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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

−11.12690156154549586391181612821, −10.23887304164718559300445456074, −9.614089741564613348442162414743, −8.418399714905025509246912298765, −7.74813796434776696248036403046, −6.44134930736606171642161345190, −5.16251389718740119205092304534, −4.21984599725299860092506647155, −3.53781121665117620597226916393, −2.28987235795038026789011060878, 0.821086080139625691166048480434, 2.34809706214048849051261345275, 3.82480788159923377483796216474, 5.34684777121050003458687731828, 6.03353842213114289625604104709, 6.73421322849436957032510519482, 7.70049717532421911844104141699, 8.800003235719382674111821966497, 9.511845461857707106720916625088, 10.96595321706806603099398311831

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