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Properties

Label 2-525-7.6-c2-0-13
Degree $2$
Conductor $525$
Sign $-0.804 - 0.593i$
Analytic cond. $14.3052$
Root an. cond. $3.78222$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.79·2-s + 1.73i·3-s + 3.79·4-s + 4.83i·6-s + (−4.15 + 5.63i)7-s − 0.578·8-s − 2.99·9-s − 18.9·11-s + 6.56i·12-s + 10.9i·13-s + (−11.6 + 15.7i)14-s − 16.7·16-s + 22.3i·17-s − 8.37·18-s − 19.6i·19-s + ⋯
L(s)  = 1  + 1.39·2-s + 0.577i·3-s + 0.948·4-s + 0.805i·6-s + (−0.593 + 0.804i)7-s − 0.0723·8-s − 0.333·9-s − 1.72·11-s + 0.547i·12-s + 0.844i·13-s + (−0.829 + 1.12i)14-s − 1.04·16-s + 1.31i·17-s − 0.465·18-s − 1.03i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.804 - 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(525\)    =    \(3 \cdot 5^{2} \cdot 7\)
Sign: $-0.804 - 0.593i$
Analytic conductor: \(14.3052\)
Root analytic conductor: \(3.78222\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{525} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 525,\ (\ :1),\ -0.804 - 0.593i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.984974504\)
\(L(\frac12)\) \(\approx\) \(1.984974504\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73iT \)
5 \( 1 \)
7 \( 1 + (4.15 - 5.63i)T \)
good2 \( 1 - 2.79T + 4T^{2} \)
11 \( 1 + 18.9T + 121T^{2} \)
13 \( 1 - 10.9iT - 169T^{2} \)
17 \( 1 - 22.3iT - 289T^{2} \)
19 \( 1 + 19.6iT - 361T^{2} \)
23 \( 1 - 31.9T + 529T^{2} \)
29 \( 1 - 39.9T + 841T^{2} \)
31 \( 1 - 36.6iT - 961T^{2} \)
37 \( 1 + 8.94T + 1.36e3T^{2} \)
41 \( 1 + 37.6iT - 1.68e3T^{2} \)
43 \( 1 - 18.8T + 1.84e3T^{2} \)
47 \( 1 - 49.3iT - 2.20e3T^{2} \)
53 \( 1 + 49.2T + 2.80e3T^{2} \)
59 \( 1 - 35.2iT - 3.48e3T^{2} \)
61 \( 1 - 63.4iT - 3.72e3T^{2} \)
67 \( 1 + 21.3T + 4.48e3T^{2} \)
71 \( 1 - 36.2T + 5.04e3T^{2} \)
73 \( 1 - 6.66iT - 5.32e3T^{2} \)
79 \( 1 + 16.2T + 6.24e3T^{2} \)
83 \( 1 - 36.7iT - 6.88e3T^{2} \)
89 \( 1 + 88.0iT - 7.92e3T^{2} \)
97 \( 1 + 133. iT - 9.40e3T^{2} \)
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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

−11.07012714401069672921951122356, −10.39208679242945921247428102557, −9.166866598161622207176763891729, −8.521483934010905914789026505585, −6.99507087853386668907996833356, −6.06250499338926099930743905725, −5.18686960375529267209368466814, −4.53581226091813833570936144463, −3.21403746034273283397698617916, −2.52775994395786625161772839399, 0.45536301390514148797426070092, 2.66268401866232404199162388803, 3.29429009007466391040912606461, 4.73086990251504432046672409958, 5.44178768901722967043329929237, 6.44556665071356611926756747260, 7.37024945179026633586192474955, 8.153863694902255387302337720236, 9.604695777002179156880998789151, 10.52645838674705619641846404733

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