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Properties

Label 2-138-23.22-c4-0-14
Degree $2$
Conductor $138$
Sign $0.268 + 0.963i$
Analytic cond. $14.2650$
Root an. cond. $3.77691$
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  + 2.82·2-s + 5.19·3-s + 8.00·4-s − 33.4i·5-s + 14.6·6-s − 19.6i·7-s + 22.6·8-s + 27·9-s − 94.6i·10-s − 54.0i·11-s + 41.5·12-s − 159.·13-s − 55.6i·14-s − 173. i·15-s + 64.0·16-s − 77.2i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.500·4-s − 1.33i·5-s + 0.408·6-s − 0.401i·7-s + 0.353·8-s + 0.333·9-s − 0.946i·10-s − 0.446i·11-s + 0.288·12-s − 0.944·13-s − 0.283i·14-s − 0.773i·15-s + 0.250·16-s − 0.267i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.268 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $0.268 + 0.963i$
Analytic conductor: \(14.2650\)
Root analytic conductor: \(3.77691\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :2),\ 0.268 + 0.963i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.53909 - 1.92727i\)
\(L(\frac12)\) \(\approx\) \(2.53909 - 1.92727i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.82T \)
3 \( 1 - 5.19T \)
23 \( 1 + (-142. - 509. i)T \)
good5 \( 1 + 33.4iT - 625T^{2} \)
7 \( 1 + 19.6iT - 2.40e3T^{2} \)
11 \( 1 + 54.0iT - 1.46e4T^{2} \)
13 \( 1 + 159.T + 2.85e4T^{2} \)
17 \( 1 + 77.2iT - 8.35e4T^{2} \)
19 \( 1 + 379. iT - 1.30e5T^{2} \)
29 \( 1 - 1.62e3T + 7.07e5T^{2} \)
31 \( 1 - 651.T + 9.23e5T^{2} \)
37 \( 1 - 974. iT - 1.87e6T^{2} \)
41 \( 1 + 171.T + 2.82e6T^{2} \)
43 \( 1 + 531. iT - 3.41e6T^{2} \)
47 \( 1 + 2.57e3T + 4.87e6T^{2} \)
53 \( 1 - 2.07e3iT - 7.89e6T^{2} \)
59 \( 1 - 8.57T + 1.21e7T^{2} \)
61 \( 1 - 1.06e3iT - 1.38e7T^{2} \)
67 \( 1 - 4.74e3iT - 2.01e7T^{2} \)
71 \( 1 - 1.38e3T + 2.54e7T^{2} \)
73 \( 1 + 2.61e3T + 2.83e7T^{2} \)
79 \( 1 + 3.92e3iT - 3.89e7T^{2} \)
83 \( 1 - 1.05e4iT - 4.74e7T^{2} \)
89 \( 1 - 1.13e4iT - 6.27e7T^{2} \)
97 \( 1 + 2.77e3iT - 8.85e7T^{2} \)
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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

−12.45778414506402843054910323757, −11.60246234914517232263189902635, −10.13896426429977145513634473693, −9.053377972971150428386998848271, −8.051554162573352828650529026546, −6.83780196923462715701281418000, −5.18947017237089447092962527971, −4.40189358154055408489984738036, −2.82493221657629260712462137673, −1.01673628048019865270155541166, 2.24618904416173389805607292927, 3.18014171371512107654814517937, 4.63834479680596330684565057170, 6.25235321680601077833156169607, 7.13769964162261319503503402956, 8.272897677731200411979282451871, 9.895215178909605868436511535606, 10.58280292875970614116189330911, 11.90565499540971913473950259657, 12.66865115403524675666688788651

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