Properties

Label 2-882-7.2-c3-0-30
Degree $2$
Conductor $882$
Sign $0.386 + 0.922i$
Analytic cond. $52.0396$
Root an. cond. $7.21385$
Motivic weight $3$
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)2-s + (−1.99 − 3.46i)4-s + (3 − 5.19i)5-s − 7.99·8-s + (−6 − 10.3i)10-s + (15 + 25.9i)11-s + 2·13-s + (−8 + 13.8i)16-s + (33 + 57.1i)17-s + (26 − 45.0i)19-s − 24·20-s + 60·22-s + (57 − 98.7i)23-s + (44.5 + 77.0i)25-s + (2 − 3.46i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.268 − 0.464i)5-s − 0.353·8-s + (−0.189 − 0.328i)10-s + (0.411 + 0.712i)11-s + 0.0426·13-s + (−0.125 + 0.216i)16-s + (0.470 + 0.815i)17-s + (0.313 − 0.543i)19-s − 0.268·20-s + 0.581·22-s + (0.516 − 0.895i)23-s + (0.355 + 0.616i)25-s + (0.0150 − 0.0261i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(52.0396\)
Root analytic conductor: \(7.21385\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :3/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.721833961\)
\(L(\frac12)\) \(\approx\) \(2.721833961\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-3 + 5.19i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-15 - 25.9i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 2T + 2.19e3T^{2} \)
17 \( 1 + (-33 - 57.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-26 + 45.0i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-57 + 98.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 72T + 2.43e4T^{2} \)
31 \( 1 + (-98 - 169. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-143 + 247. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 378T + 6.89e4T^{2} \)
43 \( 1 - 164T + 7.95e4T^{2} \)
47 \( 1 + (114 - 197. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (174 + 301. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (174 + 301. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-53 + 91.7i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (298 + 516. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 630T + 3.57e5T^{2} \)
73 \( 1 + (-521 - 902. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-44 + 76.2i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 1.44e3T + 5.71e5T^{2} \)
89 \( 1 + (-687 + 1.18e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 34T + 9.12e5T^{2} \)
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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

−9.509378862932006278287501647299, −9.089030824223987748543971069676, −7.989156998121775375580681715883, −6.92435915486380404192361273043, −5.95511027388913056515955278179, −4.99242889282001835768028142554, −4.22643940687768288615656634835, −3.08441164076010918014194814156, −1.89033874185499424632568491923, −0.845934592315256702655663330449, 0.941005237095029963353314272245, 2.64899658337377096953648270649, 3.55392066360507094528170575125, 4.66425588011873060933917611903, 5.75134536372381511610656400731, 6.30646798360923233614867252891, 7.35057479006364634303717958904, 7.991330549981919406288469274755, 9.092074987881284227973494814722, 9.726797115736370527362931270699

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