Properties

Label 2-3525-705.563-c0-0-25
Degree $2$
Conductor $3525$
Sign $0.559 + 0.828i$
Analytic cond. $1.75920$
Root an. cond. $1.32634$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.258 − 0.965i)3-s + (0.500 − 0.866i)6-s + (1.22 − 1.22i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + 1.73·14-s + 1.00·16-s + (0.707 + 0.707i)17-s + (−0.965 − 0.258i)18-s + (−1.49 − 0.866i)21-s + (−0.866 − 0.5i)24-s + (0.707 + 0.707i)27-s + 1.00i·34-s + (−0.448 − 1.67i)42-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (−0.258 − 0.965i)3-s + (0.500 − 0.866i)6-s + (1.22 − 1.22i)7-s + (0.707 − 0.707i)8-s + (−0.866 + 0.499i)9-s + 1.73·14-s + 1.00·16-s + (0.707 + 0.707i)17-s + (−0.965 − 0.258i)18-s + (−1.49 − 0.866i)21-s + (−0.866 − 0.5i)24-s + (0.707 + 0.707i)27-s + 1.00i·34-s + (−0.448 − 1.67i)42-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.559 + 0.828i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.559 + 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.013608813\)
\(L(\frac12)\) \(\approx\) \(2.013608813\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 \)
47 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
7 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
59 \( 1 + 1.73T + T^{2} \)
61 \( 1 + 2T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.73iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.137465414915758534471404603620, −7.59052314148760221478700939798, −7.28085424789496293248251234996, −6.26008606314280953467150257402, −5.83126887461700047123703398221, −4.83039341849034140712799160373, −4.39315111961188426565824943562, −3.27549092526278840409852677593, −1.72309836787225465382228339224, −1.10574205523218998757335827193, 1.69510109899488286754149811894, 2.67330172861988962525983764887, 3.35122049055856724677830405214, 4.33921201271072880209797363795, 5.02690751319798241744035848658, 5.36100674032071686781412503210, 6.28002950630289862149429453782, 7.66829594611906290970452464592, 8.177768967154453372897805329846, 8.982014593825858241837679598397

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