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Properties

Label 2-525-21.20-c1-0-39
Degree $2$
Conductor $525$
Sign $-0.0406 + 0.999i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.792i·2-s + (1.18 − 1.26i)3-s + 1.37·4-s + (−1 − 0.939i)6-s + (2 − 1.73i)7-s − 2.67i·8-s + (−0.186 − 2.99i)9-s + 2.52i·11-s + (1.62 − 1.73i)12-s + 4.10i·13-s + (−1.37 − 1.58i)14-s + 0.627·16-s − 4.37·17-s + (−2.37 + 0.147i)18-s − 3.46i·19-s + ⋯
L(s)  = 1  − 0.560i·2-s + (0.684 − 0.728i)3-s + 0.686·4-s + (−0.408 − 0.383i)6-s + (0.755 − 0.654i)7-s − 0.944i·8-s + (−0.0620 − 0.998i)9-s + 0.761i·11-s + (0.469 − 0.499i)12-s + 1.13i·13-s + (−0.366 − 0.423i)14-s + 0.156·16-s − 1.06·17-s + (−0.559 + 0.0347i)18-s − 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0406 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0406 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.0406 + 0.999i$
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: \((2,\ 525,\ (\ :1/2),\ -0.0406 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54774 - 1.61194i\)
\(L(\frac12)\) \(\approx\) \(1.54774 - 1.61194i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.18 + 1.26i)T \)
5 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good2 \( 1 + 0.792iT - 2T^{2} \)
11 \( 1 - 2.52iT - 11T^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
17 \( 1 + 4.37T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 - 8.51iT - 23T^{2} \)
29 \( 1 + 0.939iT - 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 - 6.74T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4.74T + 43T^{2} \)
47 \( 1 - 1.62T + 47T^{2} \)
53 \( 1 + 1.87iT - 53T^{2} \)
59 \( 1 + 8.74T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + 4.74T + 67T^{2} \)
71 \( 1 + 0.294iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + 2.37T + 79T^{2} \)
83 \( 1 - 17.4T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−10.84410133330009901527984626773, −9.713086676263963082757410747299, −8.967002855571990760509489487900, −7.71404109989980366446907831645, −7.12089671327371578529796418527, −6.41934550863339423358096064731, −4.66036025455502599878801756939, −3.58518614421039986683783069661, −2.23246566136237335555898934150, −1.45756697331492100396381759157, 2.15558278360804170552666146541, 3.08884870290844587796562812934, 4.57903511795801412383980605138, 5.54141190815059891047079417856, 6.41433273282958980863982395942, 7.86285366798600350287657257982, 8.243978421817344486471213112731, 9.028939748744814991564033167835, 10.38610923828762090657650993622, 10.89048467268808551671247289692

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