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Properties

Label 2-3525-705.422-c0-0-15
Degree $2$
Conductor $3525$
Sign $0.972 - 0.233i$
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 + (0.358 + 0.933i)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 + (0.738 − 0.283i)12-s + 0.556·14-s + 1.16·16-s + (−0.147 + 0.147i)17-s + (−0.0700 + 1.33i)18-s + (−0.169 + 0.379i)21-s + (−0.113 + 0.255i)24-s + (−0.891 − 0.453i)27-s + (0.232 − 0.232i)28-s + ⋯
L(s)  = 1  + (0.946 − 0.946i)2-s + (0.358 + 0.933i)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 + (0.738 − 0.283i)12-s + 0.556·14-s + 1.16·16-s + (−0.147 + 0.147i)17-s + (−0.0700 + 1.33i)18-s + (−0.169 + 0.379i)21-s + (−0.113 + 0.255i)24-s + (−0.891 − 0.453i)27-s + (0.232 − 0.232i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $0.972 - 0.233i$
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: \((2,\ 3525,\ (\ :0),\ 0.972 - 0.233i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.468208375\)
\(L(\frac12)\) \(\approx\) \(2.468208375\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.358 - 0.933i)T \)
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

−8.816335212852324934764398060870, −8.239851487101485227167470993433, −7.44347931088541538515777915344, −6.15934507328208724282951615562, −5.41387295206785634123263178777, −4.68238286602522353361327516790, −4.15756638427373100028251874180, −3.26060671984436893299229225545, −2.63783039082620866958100927674, −1.67571176803975705665393919577, 1.11706554694315763733255365883, 2.31041384211586833231253668447, 3.40631244444551118137491629897, 4.19988979808628036347905475461, 5.09388117393918156843791640037, 5.84671321273892829954592500825, 6.50449196281023709116597100366, 7.15718701122048006663740162455, 7.71433344847057936937952116967, 8.348936297838466503828960382524

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