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Properties

Label 2-234-13.12-c7-0-26
Degree $2$
Conductor $234$
Sign $0.00543 - 0.999i$
Analytic cond. $73.0980$
Root an. cond. $8.54974$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8i·2-s − 64·4-s + 319. i·5-s + 1.00e3i·7-s − 512i·8-s − 2.55e3·10-s − 7.26e3i·11-s + (7.92e3 + 43.0i)13-s − 8.06e3·14-s + 4.09e3·16-s + 2.54e4·17-s + 8.74e3i·19-s − 2.04e4i·20-s + 5.80e4·22-s + 8.35e4·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.14i·5-s + 1.11i·7-s − 0.353i·8-s − 0.807·10-s − 1.64i·11-s + (0.999 + 0.00543i)13-s − 0.785·14-s + 0.250·16-s + 1.25·17-s + 0.292i·19-s − 0.570i·20-s + 1.16·22-s + 1.43·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00543 - 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.00543 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(234\)    =    \(2 \cdot 3^{2} \cdot 13\)
Sign: $0.00543 - 0.999i$
Analytic conductor: \(73.0980\)
Root analytic conductor: \(8.54974\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{234} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 234,\ (\ :7/2),\ 0.00543 - 0.999i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.403930797\)
\(L(\frac12)\) \(\approx\) \(2.403930797\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 8iT \)
3 \( 1 \)
13 \( 1 + (-7.92e3 - 43.0i)T \)
good5 \( 1 - 319. iT - 7.81e4T^{2} \)
7 \( 1 - 1.00e3iT - 8.23e5T^{2} \)
11 \( 1 + 7.26e3iT - 1.94e7T^{2} \)
17 \( 1 - 2.54e4T + 4.10e8T^{2} \)
19 \( 1 - 8.74e3iT - 8.93e8T^{2} \)
23 \( 1 - 8.35e4T + 3.40e9T^{2} \)
29 \( 1 - 1.97e5T + 1.72e10T^{2} \)
31 \( 1 + 1.38e5iT - 2.75e10T^{2} \)
37 \( 1 + 6.18e4iT - 9.49e10T^{2} \)
41 \( 1 + 7.84e5iT - 1.94e11T^{2} \)
43 \( 1 + 4.40e5T + 2.71e11T^{2} \)
47 \( 1 + 1.80e5iT - 5.06e11T^{2} \)
53 \( 1 - 5.94e5T + 1.17e12T^{2} \)
59 \( 1 + 4.34e4iT - 2.48e12T^{2} \)
61 \( 1 - 6.07e5T + 3.14e12T^{2} \)
67 \( 1 - 3.33e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.42e6iT - 9.09e12T^{2} \)
73 \( 1 + 2.18e6iT - 1.10e13T^{2} \)
79 \( 1 + 4.16e6T + 1.92e13T^{2} \)
83 \( 1 + 4.15e5iT - 2.71e13T^{2} \)
89 \( 1 - 2.04e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.09e6iT - 8.07e13T^{2} \)
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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

−11.10902665407411634207208175630, −10.23395761856300687267369759269, −8.867042851299972856165972823947, −8.316037461288385540767041148870, −7.02568404485929809538437388466, −6.03940294262703626083750532250, −5.46747250657035413003223209217, −3.57565497296683822504921448749, −2.79334269628202235869531982837, −0.871277228110544982994515668925, 0.901690103486054421450845751483, 1.40528054018674921120832457116, 3.19533100369994420008028669067, 4.45789585787672918435148206619, 5.01926504424896846951808720293, 6.74594594810755381394232001736, 7.87358107581305892238182572710, 8.878758441016273170203119248385, 9.865302852778428016893556206146, 10.55106557234646084497744466216

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