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Properties

Label 2-3528-7.4-c1-0-46
Degree $2$
Conductor $3528$
Sign $-0.198 + 0.980i$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.70 + 2.95i)5-s + (0.414 − 0.717i)11-s − 4.24·13-s + (3.70 − 6.42i)17-s + (−3.41 − 5.91i)19-s + (−2.41 − 4.18i)23-s + (−3.32 + 5.76i)25-s − 2.82·29-s + (1.41 − 2.44i)31-s + (0.828 + 1.43i)37-s − 10.2·41-s − 11.3·43-s + (−2.24 − 3.88i)47-s + (−1 + 1.73i)53-s + 2.82·55-s + ⋯
L(s)  = 1  + (0.763 + 1.32i)5-s + (0.124 − 0.216i)11-s − 1.17·13-s + (0.899 − 1.55i)17-s + (−0.783 − 1.35i)19-s + (−0.503 − 0.871i)23-s + (−0.665 + 1.15i)25-s − 0.525·29-s + (0.254 − 0.439i)31-s + (0.136 + 0.235i)37-s − 1.59·41-s − 1.72·43-s + (−0.327 − 0.566i)47-s + (−0.137 + 0.237i)53-s + 0.381·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9428394226\)
\(L(\frac12)\) \(\approx\) \(0.9428394226\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.70 - 2.95i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.414 + 0.717i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + (-3.70 + 6.42i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.41 + 5.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.41 + 4.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.82T + 29T^{2} \)
31 \( 1 + (-1.41 + 2.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.828 - 1.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + (2.24 + 3.88i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.24 + 7.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.53 + 9.58i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.65 - 9.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + (3.87 - 6.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.82 + 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-2.87 - 4.98i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.242T + 97T^{2} \)
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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

−8.334091306494545377368408915214, −7.39726453307590416472759364443, −6.82948144999194846331253806680, −6.35815502674550520659294772878, −5.27991998933904846637083419005, −4.72029722301617911125472627716, −3.36981501605995517314469342638, −2.70552904262173191040780200503, −2.02941036351308489187669091706, −0.25352818893342314356574123631, 1.50876986858334077542941875092, 1.87070428765936812123332643004, 3.38812728785423765204973214428, 4.25695942936626791125459529104, 5.09144278192276169094622865384, 5.69664093137644652203177211540, 6.33877447802078342734577225968, 7.44606558783536825872240170137, 8.224636680107151051533349975714, 8.636711454648887225199688466027

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