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Properties

Label 2-3525-705.563-c0-0-8
Degree $2$
Conductor $3525$
Sign $0.477 + 0.878i$
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  + (−0.946 − 0.946i)2-s + (−0.933 + 0.358i)3-s + 0.790i·4-s + (1.22 + 0.544i)6-s + (0.294 − 0.294i)7-s + (−0.197 + 0.197i)8-s + (0.743 − 0.669i)9-s + (−0.283 − 0.738i)12-s − 0.556·14-s + 1.16·16-s + (0.147 + 0.147i)17-s + (−1.33 − 0.0700i)18-s + (−0.169 + 0.379i)21-s + (0.113 − 0.255i)24-s + (−0.453 + 0.891i)27-s + (0.232 + 0.232i)28-s + ⋯
L(s)  = 1  + (−0.946 − 0.946i)2-s + (−0.933 + 0.358i)3-s + 0.790i·4-s + (1.22 + 0.544i)6-s + (0.294 − 0.294i)7-s + (−0.197 + 0.197i)8-s + (0.743 − 0.669i)9-s + (−0.283 − 0.738i)12-s − 0.556·14-s + 1.16·16-s + (0.147 + 0.147i)17-s + (−1.33 − 0.0700i)18-s + (−0.169 + 0.379i)21-s + (0.113 − 0.255i)24-s + (−0.453 + 0.891i)27-s + (0.232 + 0.232i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.477 + 0.878i$
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: \((2,\ 3525,\ (\ :0),\ 0.477 + 0.878i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5329467666\)
\(L(\frac12)\) \(\approx\) \(0.5329467666\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.933 - 0.358i)T \)
5 \( 1 \)
47 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (0.946 + 0.946i)T + iT^{2} \)
7 \( 1 + (-0.294 + 0.294i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.147 - 0.147i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.831 + 0.831i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (0.437 - 0.437i)T - iT^{2} \)
59 \( 1 + 0.415T + T^{2} \)
61 \( 1 + 0.618T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 0.813iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.61iT - T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
89 \( 1 - 1.90T + T^{2} \)
97 \( 1 + (-1.34 + 1.34i)T - iT^{2} \)
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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

−9.009984353175447178920104640809, −7.969484063614529781537723335168, −7.36166869312566552328420875672, −6.26584022827056720147846203294, −5.67241804388224197663754890348, −4.73111306516030075475451795389, −3.91912281801778275093144792597, −2.89609344776220821367828439410, −1.73570081986064102178846111022, −0.77027496400294509164710637853, 0.797523592106154471875387538245, 2.02775571605240930159655298323, 3.42212756350058904863781851488, 4.64916921735157624291865722221, 5.37699113400166731883410956476, 6.19236869590257129986894119096, 6.62434522332695712469104792759, 7.54041511031675832249394477406, 7.889014838438168148158880740670, 8.760746576908419498323655055924

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