Properties

Degree $2$
Conductor $3528$
Sign $-0.997 - 0.0633i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 − 1.5i)5-s + 0.999i·8-s + (1.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + (−0.5 − 0.866i)16-s − 1.73·20-s + 0.999·22-s + (−1 + 1.73i)25-s i·29-s + (−1.5 − 0.866i)31-s + (0.866 + 0.499i)32-s + (1.49 − 0.866i)40-s + (−0.866 + 0.499i)44-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 − 1.5i)5-s + 0.999i·8-s + (1.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s + (−0.5 − 0.866i)16-s − 1.73·20-s + 0.999·22-s + (−1 + 1.73i)25-s i·29-s + (−1.5 − 0.866i)31-s + (0.866 + 0.499i)32-s + (1.49 − 0.866i)40-s + (−0.866 + 0.499i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.997 - 0.0633i$
Motivic weight: \(0\)
Character: $\chi_{3528} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.997 - 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1752182745\)
\(L(\frac12)\) \(\approx\) \(0.1752182745\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + 1.73T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−8.396882211619231081979559630112, −7.72228523602223165996687919850, −7.32164675116366659269509432260, −6.01678690582957789126432851271, −5.48225692302743293197348066355, −4.71040650775833899404681191272, −3.85595229068464594085815814716, −2.49854503131194138573829266227, −1.26662938912428378822387550457, −0.14187585088491212629746735570, 1.81512168043010021678651177499, 2.80345752791468587936824401929, 3.36216327225643675048503068124, 4.19618467312770418752699434175, 5.40325792910807685579913033669, 6.62794444712785041153857536852, 7.10077849656936229823598509566, 7.63689326303382885471010980385, 8.308759309458413769922156777222, 9.132583556003926096650784734231

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