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Properties

Label 2-3528-7.4-c1-0-34
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 + (3 − 5.19i)11-s + 6·13-s + (1 − 1.73i)17-s + (2 + 3.46i)19-s + (1 + 1.73i)23-s + (0.500 − 0.866i)25-s − 8·29-s + (2 − 3.46i)31-s + (3 + 5.19i)37-s + 10·41-s − 4·43-s + (2 + 3.46i)47-s + (−2 + 3.46i)53-s − 12·55-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + (0.904 − 1.56i)11-s + 1.66·13-s + (0.242 − 0.420i)17-s + (0.458 + 0.794i)19-s + (0.208 + 0.361i)23-s + (0.100 − 0.173i)25-s − 1.48·29-s + (0.359 − 0.622i)31-s + (0.493 + 0.854i)37-s + 1.56·41-s − 0.609·43-s + (0.291 + 0.505i)47-s + (−0.274 + 0.475i)53-s − 1.61·55-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: $\chi_{3528} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.013601874\)
\(L(\frac12)\) \(\approx\) \(2.013601874\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3 - 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-2 - 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
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 \( 1 + 83T^{2} \)
89 \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.295023896267385061275607708472, −8.029577197452804059379952354159, −6.89615936411685841124371654939, −5.86638544713334092593953757412, −5.74366097791981921822208983570, −4.39548419357018262452162177714, −3.75438122028500785591262019069, −3.08850448664679287822549028900, −1.42925876137627701846308689913, −0.75090183608207605383098160578, 1.16829104384579676334603675886, 2.20535994662372310825988280562, 3.39764029631657180243475095437, 3.94107821959265308267700049238, 4.79613993214622136243609025512, 5.90349585021183599700280251037, 6.57122699526027125603996781185, 7.27970938946927070859648463986, 7.74793813406754934169506697800, 8.990514810221194075889850625207

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