Properties

Label 2-2268-21.17-c1-0-22
Degree $2$
Conductor $2268$
Sign $0.803 + 0.595i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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  + (1.43 + 2.48i)5-s + (−0.736 − 2.54i)7-s + (2.34 + 1.35i)11-s − 3.68i·13-s + (3.22 − 5.58i)17-s + (2.73 − 1.58i)19-s + (−2.59 + 1.49i)23-s + (−1.61 + 2.79i)25-s + 2.86i·29-s + (−8.26 − 4.77i)31-s + (5.25 − 5.47i)35-s + (−1.70 − 2.95i)37-s + 1.58·41-s + 9.35·43-s + (−5.65 − 9.79i)47-s + ⋯
L(s)  = 1  + (0.641 + 1.11i)5-s + (−0.278 − 0.960i)7-s + (0.708 + 0.408i)11-s − 1.02i·13-s + (0.781 − 1.35i)17-s + (0.628 − 0.362i)19-s + (−0.540 + 0.311i)23-s + (−0.322 + 0.558i)25-s + 0.532i·29-s + (−1.48 − 0.857i)31-s + (0.888 − 0.925i)35-s + (−0.280 − 0.485i)37-s + 0.248·41-s + 1.42·43-s + (−0.824 − 1.42i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.803 + 0.595i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.803 + 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.968544843\)
\(L(\frac12)\) \(\approx\) \(1.968544843\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.736 + 2.54i)T \)
good5 \( 1 + (-1.43 - 2.48i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.34 - 1.35i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.68iT - 13T^{2} \)
17 \( 1 + (-3.22 + 5.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.73 + 1.58i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 - 1.49i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.86iT - 29T^{2} \)
31 \( 1 + (8.26 + 4.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.70 + 2.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.58T + 41T^{2} \)
43 \( 1 - 9.35T + 43T^{2} \)
47 \( 1 + (5.65 + 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.16 - 1.24i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.33 + 7.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.566 + 0.327i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.86 - 6.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.86iT - 71T^{2} \)
73 \( 1 + (-11.0 - 6.39i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.59 + 4.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 + (-3.14 - 5.45i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.2iT - 97T^{2} \)
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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

−9.325478810598909915616335137862, −7.88595277154432198276741760526, −7.26212978829365249920770300692, −6.81634081567850693334302254929, −5.83169056927189900497130379984, −5.10880361790516529173405205560, −3.80571489347433598129728401531, −3.20275646378729716985378922426, −2.16634884102049849697575294265, −0.73809717405247040542469163815, 1.28135322934874205881713402077, 2.02111303338941674428499980458, 3.40130179187864372135960098578, 4.28209360628589867556360152354, 5.31749679755662834804765307157, 5.90465102419764675057124630157, 6.49430933041358188675865541120, 7.73619677257401404267779359933, 8.559082544553587836542551049545, 9.134797747581061504521892829691

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