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Properties

Label 2-18e2-9.5-c4-0-12
Degree $2$
Conductor $324$
Sign $-0.342 + 0.939i$
Analytic cond. $33.4918$
Root an. cond. $5.78721$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.79 + 4.5i)5-s + (−2.5 − 4.33i)7-s + (−101. + 58.5i)11-s + (17 − 29.4i)13-s − 450i·17-s − 64·19-s + (530. + 306i)23-s + (−272 − 471. i)25-s + (−919. + 531i)29-s + (348.5 − 603. i)31-s − 45.0i·35-s − 748·37-s + (592. + 342i)41-s + (−1.30e3 − 2.26e3i)43-s + (2.29e3 − 1.32e3i)47-s + ⋯
L(s)  = 1  + (0.311 + 0.179i)5-s + (−0.0510 − 0.0883i)7-s + (−0.837 + 0.483i)11-s + (0.100 − 0.174i)13-s − 1.55i·17-s − 0.177·19-s + (1.00 + 0.578i)23-s + (−0.435 − 0.753i)25-s + (−1.09 + 0.631i)29-s + (0.362 − 0.628i)31-s − 0.0367i·35-s − 0.546·37-s + (0.352 + 0.203i)41-s + (−0.707 − 1.22i)43-s + (1.03 − 0.598i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(33.4918\)
Root analytic conductor: \(5.78721\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :2),\ -0.342 + 0.939i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.100886732\)
\(L(\frac12)\) \(\approx\) \(1.100886732\)
\(L(3)\) 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 \( 1 \)
good5 \( 1 + (-7.79 - 4.5i)T + (312.5 + 541. i)T^{2} \)
7 \( 1 + (2.5 + 4.33i)T + (-1.20e3 + 2.07e3i)T^{2} \)
11 \( 1 + (101. - 58.5i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (-17 + 29.4i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 450iT - 8.35e4T^{2} \)
19 \( 1 + 64T + 1.30e5T^{2} \)
23 \( 1 + (-530. - 306i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (919. - 531i)T + (3.53e5 - 6.12e5i)T^{2} \)
31 \( 1 + (-348.5 + 603. i)T + (-4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + 748T + 1.87e6T^{2} \)
41 \( 1 + (-592. - 342i)T + (1.41e6 + 2.44e6i)T^{2} \)
43 \( 1 + (1.30e3 + 2.26e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-2.29e3 + 1.32e3i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 1.07e3iT - 7.89e6T^{2} \)
59 \( 1 + (5.03e3 + 2.90e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (3.20e3 + 5.54e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-2.60e3 + 4.51e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 6.57e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.51e3T + 2.83e7T^{2} \)
79 \( 1 + (3.75e3 + 6.49e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + (-4.74e3 + 2.74e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 8.87e3iT - 6.27e7T^{2} \)
97 \( 1 + (5.28e3 + 9.15e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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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

−10.64322008180985428352173707974, −9.794873088592668214886150404632, −8.937384902908502837959565139351, −7.66823073958378999827340186035, −6.95844107678543247448711212068, −5.63793315064998446251775143456, −4.76114503288444828360122481648, −3.24945992923008744686857057921, −2.08098374861850014837142379707, −0.32439862576861860275043806359, 1.41225581314495066999077172302, 2.82110133782916402716509672270, 4.15293334586567211237677700509, 5.44240297028065295654696892217, 6.24291170867259545871993361866, 7.53023605800599133112164574034, 8.483250715616078227439003459923, 9.327113079496955135831507981125, 10.49136850057957211483638509500, 11.04180171563725793623889847633

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