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Properties

Label 2-3528-392.285-c0-0-0
Degree $2$
Conductor $3528$
Sign $0.918 - 0.394i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
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  + (−0.680 + 0.733i)2-s + (−0.0747 − 0.997i)4-s + (0.215 + 0.548i)5-s + (0.365 − 0.930i)7-s + (0.781 + 0.623i)8-s + (−0.548 − 0.215i)10-s + (−0.0440 − 0.142i)11-s + (0.433 + 0.900i)14-s + (−0.988 + 0.149i)16-s + (0.531 − 0.255i)20-s + (0.134 + 0.0648i)22-s + (0.478 − 0.443i)25-s + (−0.955 − 0.294i)28-s + (0.636 + 1.32i)29-s + (0.258 − 0.149i)31-s + (0.563 − 0.826i)32-s + ⋯
L(s)  = 1  + (−0.680 + 0.733i)2-s + (−0.0747 − 0.997i)4-s + (0.215 + 0.548i)5-s + (0.365 − 0.930i)7-s + (0.781 + 0.623i)8-s + (−0.548 − 0.215i)10-s + (−0.0440 − 0.142i)11-s + (0.433 + 0.900i)14-s + (−0.988 + 0.149i)16-s + (0.531 − 0.255i)20-s + (0.134 + 0.0648i)22-s + (0.478 − 0.443i)25-s + (−0.955 − 0.294i)28-s + (0.636 + 1.32i)29-s + (0.258 − 0.149i)31-s + (0.563 − 0.826i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.918 - 0.394i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.918 - 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9797121777\)
\(L(\frac12)\) \(\approx\) \(0.9797121777\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.680 - 0.733i)T \)
3 \( 1 \)
7 \( 1 + (-0.365 + 0.930i)T \)
good5 \( 1 + (-0.215 - 0.548i)T + (-0.733 + 0.680i)T^{2} \)
11 \( 1 + (0.0440 + 0.142i)T + (-0.826 + 0.563i)T^{2} \)
13 \( 1 + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + (-0.365 - 0.930i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.365 - 0.930i)T^{2} \)
29 \( 1 + (-0.636 - 1.32i)T + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.258 + 0.149i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.988 + 0.149i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.0747 - 0.997i)T^{2} \)
53 \( 1 + (-0.997 + 0.0747i)T + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (-0.632 + 1.61i)T + (-0.733 - 0.680i)T^{2} \)
61 \( 1 + (-0.988 - 0.149i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-1.06 - 1.14i)T + (-0.0747 + 0.997i)T^{2} \)
79 \( 1 + (-0.365 + 0.632i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.302 + 1.32i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.826 - 0.563i)T^{2} \)
97 \( 1 + 1.12iT - T^{2} \)
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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

−8.551081311843444993599215968770, −8.140527018498892081214648684404, −7.11228267336908036223726919245, −6.88884125471697972943966811042, −6.01292637902843990045945674259, −5.11537568910777389386625158152, −4.40114032977476371213416523762, −3.27737401338072358382160836443, −2.07141953961756806862336341348, −0.918925978638913004094422286160, 1.08361669391317668251335068883, 2.14645358895411462937152356972, 2.84056959307049818837339614257, 3.99249984324832571062303243937, 4.81064038957791995937221243390, 5.59421446495634181764668834944, 6.56697697183649428290780610481, 7.48498478218269558810885358470, 8.288618330613518248321011444940, 8.702419745828437595717070990739

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