Properties

Label 2-234-13.3-c3-0-14
Degree $2$
Conductor $234$
Sign $-0.999 + 0.0256i$
Analytic cond. $13.8064$
Root an. cond. $3.71570$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s − 2·5-s + (2.5 − 4.33i)7-s + 7.99·8-s + (2 + 3.46i)10-s + (6.5 + 11.2i)11-s + (−13 − 45.0i)13-s − 10·14-s + (−8 − 13.8i)16-s + (13.5 − 23.3i)17-s + (−37.5 + 64.9i)19-s + (3.99 − 6.92i)20-s + (12.9 − 22.5i)22-s + (−93.5 − 161. i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.178·5-s + (0.134 − 0.233i)7-s + 0.353·8-s + (0.0632 + 0.109i)10-s + (0.178 + 0.308i)11-s + (−0.277 − 0.960i)13-s − 0.190·14-s + (−0.125 − 0.216i)16-s + (0.192 − 0.333i)17-s + (−0.452 + 0.784i)19-s + (0.0447 − 0.0774i)20-s + (0.125 − 0.218i)22-s + (−0.847 − 1.46i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $-0.999 + 0.0256i$
Analytic conductor: \(13.8064\)
Root analytic conductor: \(3.71570\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :3/2),\ -0.999 + 0.0256i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00678483 - 0.529057i\)
\(L(\frac12)\) \(\approx\) \(0.00678483 - 0.529057i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
3 \( 1 \)
13 \( 1 + (13 + 45.0i)T \)
good5 \( 1 + 2T + 125T^{2} \)
7 \( 1 + (-2.5 + 4.33i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-6.5 - 11.2i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-13.5 + 23.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (37.5 - 64.9i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (93.5 + 161. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (6.5 + 11.2i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 104T + 2.97e4T^{2} \)
37 \( 1 + (211.5 + 366. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-97.5 - 168. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (99.5 - 172. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + 388T + 1.03e5T^{2} \)
53 \( 1 + 618T + 1.48e5T^{2} \)
59 \( 1 + (-245.5 + 425. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (87.5 - 151. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (408.5 + 707. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-39.5 + 68.4i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 230T + 3.89e5T^{2} \)
79 \( 1 - 764T + 4.93e5T^{2} \)
83 \( 1 - 732T + 5.71e5T^{2} \)
89 \( 1 + (520.5 + 901. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-48.5 + 84.0i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−11.13243683566525506202112744939, −10.32447045042845300226745316014, −9.504387657461958687740621762661, −8.253465538585950322023791900578, −7.55819507344574130355609432726, −6.11249118499857557139992291966, −4.66774335773028108652200902596, −3.47862707422115986008938417936, −1.98748326894650953540629312077, −0.23470742318623410524152968593, 1.79355807716062407327090008259, 3.75968138768105689831044744269, 5.08703164125113001936945030048, 6.22915333084335052406297264582, 7.23885943210726294913719967601, 8.264302430068578243975355485934, 9.177662856565663163725073255431, 10.06131008647835730763082470652, 11.30968155635448641925416233685, 12.01201529798793982259223724797

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