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Properties

Label 2-1323-49.45-c0-0-0
Degree $2$
Conductor $1323$
Sign $0.304 + 0.952i$
Analytic cond. $0.660263$
Root an. cond. $0.812565$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.365 − 0.930i)4-s + (0.900 − 0.433i)7-s + (−0.255 − 0.531i)13-s + (−0.733 + 0.680i)16-s + (0.751 − 0.433i)19-s + (0.826 − 0.563i)25-s + (−0.733 − 0.680i)28-s + (−1.61 − 0.930i)31-s + (−0.266 + 0.680i)37-s + (0.0332 + 0.145i)43-s + (0.623 − 0.781i)49-s + (−0.400 + 0.432i)52-s + (0.277 + 0.108i)61-s + (0.900 + 0.433i)64-s + (0.733 − 1.26i)67-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)4-s + (0.900 − 0.433i)7-s + (−0.255 − 0.531i)13-s + (−0.733 + 0.680i)16-s + (0.751 − 0.433i)19-s + (0.826 − 0.563i)25-s + (−0.733 − 0.680i)28-s + (−1.61 − 0.930i)31-s + (−0.266 + 0.680i)37-s + (0.0332 + 0.145i)43-s + (0.623 − 0.781i)49-s + (−0.400 + 0.432i)52-s + (0.277 + 0.108i)61-s + (0.900 + 0.433i)64-s + (0.733 − 1.26i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.304 + 0.952i$
Analytic conductor: \(0.660263\)
Root analytic conductor: \(0.812565\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1270, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :0),\ 0.304 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.046968824\)
\(L(\frac12)\) \(\approx\) \(1.046968824\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (-0.900 + 0.433i)T \)
good2 \( 1 + (0.365 + 0.930i)T^{2} \)
5 \( 1 + (-0.826 + 0.563i)T^{2} \)
11 \( 1 + (-0.988 - 0.149i)T^{2} \)
13 \( 1 + (0.255 + 0.531i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (-0.751 + 0.433i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.955 + 0.294i)T^{2} \)
29 \( 1 + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (1.61 + 0.930i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.266 - 0.680i)T + (-0.733 - 0.680i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (-0.0332 - 0.145i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.365 - 0.930i)T^{2} \)
53 \( 1 + (-0.733 + 0.680i)T^{2} \)
59 \( 1 + (-0.826 - 0.563i)T^{2} \)
61 \( 1 + (-0.277 - 0.108i)T + (0.733 + 0.680i)T^{2} \)
67 \( 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.880 + 1.29i)T + (-0.365 + 0.930i)T^{2} \)
79 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.988 - 0.149i)T^{2} \)
97 \( 1 - 1.99iT - T^{2} \)
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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

−9.675838861468718591496573939771, −8.966334859456534613954685250843, −8.049383399088252810255657580691, −7.26824637387688242128031184313, −6.29094244363363328176405738027, −5.26091075396337235046442407216, −4.81922782634146292338930599664, −3.70062011242145264556648357203, −2.23076976950991820046287219988, −0.983877278254931319822383704258, 1.75252778204371813879674385188, 2.98790083069800652164428781202, 3.96276922140315039872895976016, 4.91148072648213232272629511347, 5.62008423436526011214105108723, 7.06519947997700995730717498070, 7.49355768411657589788505623737, 8.535699455811223537970473250494, 8.932848126999100481305464682994, 9.823462040418878059960022066535

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