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Properties

Label 2-3528-7.2-c1-0-36
Degree $2$
Conductor $3528$
Sign $-0.266 + 0.963i$
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 + 1.73i)5-s + 2·13-s + (−3 − 5.19i)17-s + (−2 + 3.46i)19-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s − 6·29-s + (−4 − 6.92i)31-s + (5 − 8.66i)37-s − 10·41-s + 12·43-s + (4 − 6.92i)47-s + (3 + 5.19i)53-s + (−2 − 3.46i)59-s + (−5 + 8.66i)61-s + ⋯
L(s)  = 1  + (−0.447 + 0.774i)5-s + 0.554·13-s + (−0.727 − 1.26i)17-s + (−0.458 + 0.794i)19-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s − 1.11·29-s + (−0.718 − 1.24i)31-s + (0.821 − 1.42i)37-s − 1.56·41-s + 1.82·43-s + (0.583 − 1.01i)47-s + (0.412 + 0.713i)53-s + (−0.260 − 0.450i)59-s + (−0.640 + 1.10i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.266 + 0.963i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6333828253\)
\(L(\frac12)\) \(\approx\) \(0.6333828253\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.287473194015390875199421027266, −7.32114585838750228813597999279, −7.21933097359971682697723695376, −6.01150472475383942470278631495, −5.54441547913117532361085288489, −4.26022168705495187138359738218, −3.73910200655359719564867049375, −2.76218814417333193748341307604, −1.81661694810478825367068465998, −0.19865364056486288441435363131, 1.15935882014592665747642053826, 2.22615830680149792870735143565, 3.42240605440209132099647609503, 4.29369951988747574389436800426, 4.78518800285213778716035309575, 5.87741474396597399751439054251, 6.48365262117789232112006071306, 7.36321481715390707455473010977, 8.271751151504484456256315239142, 8.684550184011888664313147393708

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