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Properties

Label 2-525-5.4-c5-0-58
Degree $2$
Conductor $525$
Sign $0.894 + 0.447i$
Analytic cond. $84.2015$
Root an. cond. $9.17613$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 10.8i·2-s − 9i·3-s − 86.5·4-s + 97.9·6-s − 49i·7-s − 593. i·8-s − 81·9-s + 210.·11-s + 778. i·12-s + 407. i·13-s + 533.·14-s + 3.69e3·16-s + 1.71e3i·17-s − 881. i·18-s − 2.27e3·19-s + ⋯
L(s)  = 1  + 1.92i·2-s − 0.577i·3-s − 2.70·4-s + 1.11·6-s − 0.377i·7-s − 3.27i·8-s − 0.333·9-s + 0.524·11-s + 1.56i·12-s + 0.668i·13-s + 0.727·14-s + 3.60·16-s + 1.44i·17-s − 0.641i·18-s − 1.44·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(84.2015\)
Root analytic conductor: \(9.17613\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :5/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.3496905633\)
\(L(\frac12)\) \(\approx\) \(0.3496905633\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 9iT \)
5 \( 1 \)
7 \( 1 + 49iT \)
good2 \( 1 - 10.8iT - 32T^{2} \)
11 \( 1 - 210.T + 1.61e5T^{2} \)
13 \( 1 - 407. iT - 3.71e5T^{2} \)
17 \( 1 - 1.71e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.27e3T + 2.47e6T^{2} \)
23 \( 1 - 2.08e3iT - 6.43e6T^{2} \)
29 \( 1 - 6.77e3T + 2.05e7T^{2} \)
31 \( 1 + 2.39e3T + 2.86e7T^{2} \)
37 \( 1 - 1.07e4iT - 6.93e7T^{2} \)
41 \( 1 - 3.12e3T + 1.15e8T^{2} \)
43 \( 1 + 1.58e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.47e4iT - 2.29e8T^{2} \)
53 \( 1 + 3.15e4iT - 4.18e8T^{2} \)
59 \( 1 + 1.21e4T + 7.14e8T^{2} \)
61 \( 1 + 4.21e4T + 8.44e8T^{2} \)
67 \( 1 + 2.88e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.33e4T + 1.80e9T^{2} \)
73 \( 1 - 8.38e4iT - 2.07e9T^{2} \)
79 \( 1 + 3.48e4T + 3.07e9T^{2} \)
83 \( 1 - 3.51e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.41e4T + 5.58e9T^{2} \)
97 \( 1 + 1.07e5iT - 8.58e9T^{2} \)
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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

−9.615155339971917776396185083361, −8.531144460342171147637737128614, −8.229661473069222660366657024363, −6.99332281001484466776930298513, −6.58067071272510034771563148023, −5.76209555230298342575239956795, −4.55932271365296399529310978734, −3.74510299470061634932788538321, −1.52873716899968019658569405582, −0.098653564951761254446302196143, 0.969045397316392129789027464847, 2.39920281271969837659374616580, 3.05727265525494687638848156056, 4.27769597642656433406996765142, 4.85603883257604903074949169706, 6.11215393724755171468879813131, 7.938843818268713554630351313690, 8.997488736783336446144903151196, 9.331034570477110174440775317119, 10.47652609044955047916325544520

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