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Properties

Label 2-242-11.4-c3-0-12
Degree $2$
Conductor $242$
Sign $0.350 - 0.936i$
Analytic cond. $14.2784$
Root an. cond. $3.77868$
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  + (1.61 + 1.17i)2-s + (0.309 − 0.951i)3-s + (1.23 + 3.80i)4-s + (2.42 − 1.76i)5-s + (1.61 − 1.17i)6-s + (3.09 + 9.51i)7-s + (−2.47 + 7.60i)8-s + (21.0 + 15.2i)9-s + 6·10-s + 4·12-s + (−12.9 − 9.40i)13-s + (−6.18 + 19.0i)14-s + (−0.927 − 2.85i)15-s + (−12.9 + 9.40i)16-s + (33.9 − 24.6i)17-s + (16.0 + 49.4i)18-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (0.0594 − 0.183i)3-s + (0.154 + 0.475i)4-s + (0.217 − 0.157i)5-s + (0.110 − 0.0799i)6-s + (0.166 + 0.513i)7-s + (−0.109 + 0.336i)8-s + (0.779 + 0.566i)9-s + 0.189·10-s + 0.0962·12-s + (−0.276 − 0.200i)13-s + (−0.117 + 0.363i)14-s + (−0.0159 − 0.0491i)15-s + (−0.202 + 0.146i)16-s + (0.484 − 0.352i)17-s + (0.210 + 0.647i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 242 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(242\)    =    \(2 \cdot 11^{2}\)
Sign: $0.350 - 0.936i$
Analytic conductor: \(14.2784\)
Root analytic conductor: \(3.77868\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{242} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 242,\ (\ :3/2),\ 0.350 - 0.936i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.24102 + 1.55409i\)
\(L(\frac12)\) \(\approx\) \(2.24102 + 1.55409i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.61 - 1.17i)T \)
11 \( 1 \)
good3 \( 1 + (-0.309 + 0.951i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (-2.42 + 1.76i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (-3.09 - 9.51i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (12.9 + 9.40i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-33.9 + 24.6i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (35.8 - 110. i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 - 189T + 1.21e4T^{2} \)
29 \( 1 + (-37.0 - 114. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (-131. - 95.8i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (126. + 388. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (144. - 445. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 + 110T + 7.95e4T^{2} \)
47 \( 1 + (-44.4 + 136. i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (72.8 + 52.9i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (139. + 430. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (-16.1 + 11.7i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + 97T + 3.00e5T^{2} \)
71 \( 1 + (-376. + 273. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (262. + 806. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (600. + 436. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-354. + 257. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 + 273T + 7.04e5T^{2} \)
97 \( 1 + (615. + 447. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−12.19652475568443430724414816833, −10.98776520036502139140626690907, −9.939428132602979706863644218898, −8.761932724035732789917529222105, −7.73912589462974467253921488226, −6.84576544543074493346820133560, −5.55262394048510504959925556473, −4.75609033959007399414907475048, −3.23058304907260090890264373972, −1.68005506874555467896971630588, 1.01043894114988702933199466911, 2.69069110948681975316449882455, 4.05332170259779945509188410129, 4.93825692112395128129644809296, 6.42858574098821595203562880039, 7.21699187663479455075156068327, 8.734591384484999603448627215003, 9.847977080551366546311893696465, 10.51097701686077733203796365155, 11.53614867686051212003808466439

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