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Properties

Label 1-1323-1323.5-r0-0-0
Degree $1$
Conductor $1323$
Sign $0.111 - 0.993i$
Analytic cond. $6.14398$
Root an. cond. $6.14398$
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  + (0.124 + 0.992i)2-s + (−0.969 + 0.246i)4-s + (−0.853 + 0.521i)5-s + (−0.365 − 0.930i)8-s + (−0.623 − 0.781i)10-s + (0.124 + 0.992i)11-s + (0.998 + 0.0498i)13-s + (0.878 − 0.478i)16-s + (−0.900 + 0.433i)17-s − 19-s + (0.698 − 0.715i)20-s + (−0.969 + 0.246i)22-s + (−0.270 + 0.962i)23-s + (0.456 − 0.889i)25-s + (0.0747 + 0.997i)26-s + ⋯
L(s)  = 1  + (0.124 + 0.992i)2-s + (−0.969 + 0.246i)4-s + (−0.853 + 0.521i)5-s + (−0.365 − 0.930i)8-s + (−0.623 − 0.781i)10-s + (0.124 + 0.992i)11-s + (0.998 + 0.0498i)13-s + (0.878 − 0.478i)16-s + (−0.900 + 0.433i)17-s − 19-s + (0.698 − 0.715i)20-s + (−0.969 + 0.246i)22-s + (−0.270 + 0.962i)23-s + (0.456 − 0.889i)25-s + (0.0747 + 0.997i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.111 - 0.993i$
Analytic conductor: \(6.14398\)
Root analytic conductor: \(6.14398\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1323,\ (0:\ ),\ 0.111 - 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1480137310 + 0.1323521987i\)
\(L(\frac12)\) \(\approx\) \(-0.1480137310 + 0.1323521987i\)
\(L(1)\) \(\approx\) \(0.5300256935 + 0.4651643113i\)
\(L(1)\) \(\approx\) \(0.5300256935 + 0.4651643113i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.124 + 0.992i)T \)
5 \( 1 + (-0.853 + 0.521i)T \)
11 \( 1 + (0.124 + 0.992i)T \)
13 \( 1 + (0.998 + 0.0498i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.270 + 0.962i)T \)
29 \( 1 + (-0.270 - 0.962i)T \)
31 \( 1 + (-0.766 + 0.642i)T \)
37 \( 1 + (0.0747 - 0.997i)T \)
41 \( 1 + (0.878 + 0.478i)T \)
43 \( 1 + (-0.0249 + 0.999i)T \)
47 \( 1 + (-0.797 + 0.603i)T \)
53 \( 1 + (-0.826 + 0.563i)T \)
59 \( 1 + (-0.661 + 0.749i)T \)
61 \( 1 + (0.969 + 0.246i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (-0.0747 - 0.997i)T \)
73 \( 1 + (0.733 - 0.680i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.998 + 0.0498i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.37683628145022432729612906737, −19.7403628822226175765028999669, −18.81640401597891682662646910972, −18.52833981545905252530386468654, −17.400107843027528245129806793967, −16.53339016910325348803296348685, −15.83599434506080445175058618442, −14.87959285954055745951377260852, −14.04513380720021946700892289944, −13.12394510339206262589029017029, −12.72184099583961649801485926482, −11.65981097801010430012929042126, −11.13008725388483381621404367731, −10.56434079560126149213544252759, −9.28698121349749234598779083630, −8.581050898580653159456133895134, −8.22088230645997381460954444293, −6.790576896526100300529895436460, −5.78006907399681631304732004829, −4.80711678033640206318496875656, −3.99263434896046828884761388805, −3.39236855179379255441551537131, −2.25991039193410230212937021641, −1.1190591804629653208243708470, −0.084236905706805563611721047103, 1.67300868809827846762861164953, 3.094196062131061614500322750510, 4.15621880237285134524217038036, 4.42491433251233873189386557745, 5.84993541485759727965955552036, 6.490298701380542353493584462028, 7.322859129214381686646739188785, 7.96682658518345128666410459084, 8.80055611102624507819814840477, 9.61616289001644671780488592433, 10.70976994056145692466244643259, 11.42139079540999743581604509913, 12.54387648169239496271251624041, 13.068923007856109843907623675110, 14.09260273195644771777203332715, 14.84516542000293555043065240216, 15.42605638587126194784633196142, 15.964395098483602182665024186141, 16.86286965403982068232242892028, 17.88158632638468254695786858751, 18.10302281591129725831655925427, 19.31091117888907203617124030263, 19.68086092705347299823125717150, 20.91506377588261887917069050760, 21.68890510276507965379915112272

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