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Properties

Label 2-2268-21.17-c1-0-16
Degree $2$
Conductor $2268$
Sign $-0.0647 - 0.997i$
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  + (1.76 + 3.06i)5-s + (2.20 + 1.46i)7-s + (3.01 + 1.74i)11-s + 6.98i·13-s + (0.128 − 0.223i)17-s + (3.18 − 1.83i)19-s + (3.40 − 1.96i)23-s + (−3.76 + 6.52i)25-s − 0.0544i·29-s + (−4.84 − 2.79i)31-s + (−0.579 + 9.34i)35-s + (−1.66 − 2.88i)37-s + 10.1·41-s − 6.57·43-s + (−6.19 − 10.7i)47-s + ⋯
L(s)  = 1  + (0.791 + 1.37i)5-s + (0.833 + 0.552i)7-s + (0.908 + 0.524i)11-s + 1.93i·13-s + (0.0312 − 0.0541i)17-s + (0.729 − 0.421i)19-s + (0.710 − 0.410i)23-s + (−0.753 + 1.30i)25-s − 0.0101i·29-s + (−0.870 − 0.502i)31-s + (−0.0979 + 1.58i)35-s + (−0.273 − 0.474i)37-s + 1.58·41-s − 1.00·43-s + (−0.903 − 1.56i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0647 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0647 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.0647 - 0.997i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.0647 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.499598500\)
\(L(\frac12)\) \(\approx\) \(2.499598500\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 - 1.46i)T \)
good5 \( 1 + (-1.76 - 3.06i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.01 - 1.74i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.98iT - 13T^{2} \)
17 \( 1 + (-0.128 + 0.223i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.18 + 1.83i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.40 + 1.96i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.0544iT - 29T^{2} \)
31 \( 1 + (4.84 + 2.79i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.66 + 2.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 6.57T + 43T^{2} \)
47 \( 1 + (6.19 + 10.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.94 + 4.00i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.67 + 6.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.78 + 1.03i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.728 + 1.26i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.9iT - 71T^{2} \)
73 \( 1 + (1.58 + 0.913i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.92 - 11.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 0.314T + 83T^{2} \)
89 \( 1 + (4.84 + 8.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.5iT - 97T^{2} \)
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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

−9.402372957974960783210430774707, −8.622123821970891137509271430349, −7.40958147158947122368626363966, −6.83493214107475238139358046131, −6.32316432359861580726160846080, −5.30049155563564918333656535817, −4.42601754309530546209303125265, −3.39804926135474143223781609221, −2.21443148157293245938010116843, −1.73050654554992245165095115295, 0.974215979439683296095640512272, 1.42141833970512010958296080680, 3.00516346066930187293964230395, 4.02277417494238283082441482336, 5.09843388764667804999074533618, 5.41396392113732388511663164972, 6.28971877540568510470809223108, 7.54154658698321728529383561500, 8.084798733387118826375243106234, 8.839606241695775866505391852831

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