Properties

Label 2-3525-1.1-c1-0-35
Degree $2$
Conductor $3525$
Sign $1$
Analytic cond. $28.1472$
Root an. cond. $5.30539$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s + 2·6-s + 3·7-s + 9-s + 11-s − 2·12-s + 2·13-s − 6·14-s − 4·16-s − 2·17-s − 2·18-s + 6·19-s − 3·21-s − 2·22-s − 3·23-s − 4·26-s − 27-s + 6·28-s + 3·29-s + 2·31-s + 8·32-s − 33-s + 4·34-s + 2·36-s + 7·37-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 0.554·13-s − 1.60·14-s − 16-s − 0.485·17-s − 0.471·18-s + 1.37·19-s − 0.654·21-s − 0.426·22-s − 0.625·23-s − 0.784·26-s − 0.192·27-s + 1.13·28-s + 0.557·29-s + 0.359·31-s + 1.41·32-s − 0.174·33-s + 0.685·34-s + 1/3·36-s + 1.15·37-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $1$
Analytic conductor: \(28.1472\)
Root analytic conductor: \(5.30539\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3525} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9425185637\)
\(L(\frac12)\) \(\approx\) \(0.9425185637\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 \)
47 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 17 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−8.602584629951561113830733197974, −7.78091108170505847598252560033, −7.51664253316875775477338396262, −6.47471788578892689826689313513, −5.74730307755447516496194871617, −4.73279478889276445796998568082, −4.11797623123414630335873886332, −2.60023057208825044992396591035, −1.49548225835488390565202219199, −0.818394894647382605599021148107, 0.818394894647382605599021148107, 1.49548225835488390565202219199, 2.60023057208825044992396591035, 4.11797623123414630335873886332, 4.73279478889276445796998568082, 5.74730307755447516496194871617, 6.47471788578892689826689313513, 7.51664253316875775477338396262, 7.78091108170505847598252560033, 8.602584629951561113830733197974

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