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Properties

Label 2-3525-705.563-c0-0-17
Degree $2$
Conductor $3525$
Sign $-0.923 + 0.382i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-0.923 + 0.382i$
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: \((2,\ 3525,\ (\ :0),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03507408012\)
\(L(\frac12)\) \(\approx\) \(0.03507408012\)
\(L(1)\) 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

−8.846969264140072635669187092976, −8.179429602017002404845844061773, −6.89003689233588603859218935972, −5.99219496448879642115090565427, −5.46292908963856538781424194671, −4.56262717529636930195847749211, −3.33003097663824771469702310679, −2.80023084825287963260119746084, −2.01320590940827448571370620370, −0.02545428596973558925332537796, 1.26335389290067736961625803499, 2.75964135765979437261334562891, 3.49797638240591585257606217050, 4.31076765793652976136743215103, 5.89274220724448014794838074822, 6.45803739424543310774864200155, 6.90632470450975250288647327239, 7.63393760348121085389834375638, 8.052476914798070278362475642472, 9.090903519304288124239193319565

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