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Properties

Label 2-3525-705.422-c0-0-19
Degree $2$
Conductor $3525$
Sign $0.522 + 0.852i$
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  + (−1.38 + 1.38i)2-s + (−0.777 + 0.629i)3-s − 2.82i·4-s + (0.204 − 1.94i)6-s + (−1.05 − 1.05i)7-s + (2.52 + 2.52i)8-s + (0.207 − 0.978i)9-s + (1.77 + 2.19i)12-s + 2.90·14-s − 4.16·16-s + (1.29 − 1.29i)17-s + (1.06 + 1.64i)18-s + (1.47 + 0.155i)21-s + (−3.55 − 0.373i)24-s + (0.453 + 0.891i)27-s + (−2.97 + 2.97i)28-s + ⋯
L(s)  = 1  + (−1.38 + 1.38i)2-s + (−0.777 + 0.629i)3-s − 2.82i·4-s + (0.204 − 1.94i)6-s + (−1.05 − 1.05i)7-s + (2.52 + 2.52i)8-s + (0.207 − 0.978i)9-s + (1.77 + 2.19i)12-s + 2.90·14-s − 4.16·16-s + (1.29 − 1.29i)17-s + (1.06 + 1.64i)18-s + (1.47 + 0.155i)21-s + (−3.55 − 0.373i)24-s + (0.453 + 0.891i)27-s + (−2.97 + 2.97i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(1.75920\)
Root analytic conductor: \(1.32634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3525} (1832, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :0),\ 0.522 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1905114071\)
\(L(\frac12)\) \(\approx\) \(0.1905114071\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.777 - 0.629i)T \)
5 \( 1 \)
47 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.38 - 1.38i)T - iT^{2} \)
7 \( 1 + (1.05 + 1.05i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-1.29 + 1.29i)T - iT^{2} \)
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 \( 1 + 1.48T + T^{2} \)
61 \( 1 + 0.618T + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 - 1.98iT - T^{2} \)
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

−8.766204558718882929136327344501, −7.56322840787860278088509786199, −7.28297489270515004502888854540, −6.51898722668144081160483527587, −5.89628956485939500099367226088, −5.19365034375639515894225539279, −4.36949422796466591884621489642, −3.18086150522708196413884284736, −1.23388347317508550450344671230, −0.23821286751120918518821922467, 1.27011754978361545487989563046, 2.11022110339706344593526144093, 3.04207001367121479045832094239, 3.75040531811593093919441089776, 5.09483657383069422335349943617, 6.14473117907026326548368112545, 6.75807664891189195910565846162, 7.75921852072537048086317080881, 8.228395085450095218045037731553, 9.014338630200753268399686825691

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