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Properties

Label 2-2e7-32.21-c3-0-10
Degree $2$
Conductor $128$
Sign $-0.489 + 0.872i$
Analytic cond. $7.55224$
Root an. cond. $2.74813$
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  + (1.36 − 0.564i)3-s + (6.58 − 15.8i)5-s + (−14.5 − 14.5i)7-s + (−17.5 + 17.5i)9-s + (−34.4 − 14.2i)11-s + (15.3 + 37.1i)13-s − 25.3i·15-s − 103. i·17-s + (−12.4 − 29.9i)19-s + (−28.0 − 11.6i)21-s + (72.3 − 72.3i)23-s + (−120. − 120. i)25-s + (−29.2 + 70.5i)27-s + (23.9 − 9.90i)29-s + 124.·31-s + ⋯
L(s)  = 1  + (0.262 − 0.108i)3-s + (0.588 − 1.42i)5-s + (−0.785 − 0.785i)7-s + (−0.650 + 0.650i)9-s + (−0.944 − 0.391i)11-s + (0.328 + 0.792i)13-s − 0.436i·15-s − 1.47i·17-s + (−0.149 − 0.361i)19-s + (−0.291 − 0.120i)21-s + (0.656 − 0.656i)23-s + (−0.966 − 0.966i)25-s + (−0.208 + 0.503i)27-s + (0.153 − 0.0634i)29-s + 0.722·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.489 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $-0.489 + 0.872i$
Analytic conductor: \(7.55224\)
Root analytic conductor: \(2.74813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 128,\ (\ :3/2),\ -0.489 + 0.872i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.681586 - 1.16422i\)
\(L(\frac12)\) \(\approx\) \(0.681586 - 1.16422i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-1.36 + 0.564i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (-6.58 + 15.8i)T + (-88.3 - 88.3i)T^{2} \)
7 \( 1 + (14.5 + 14.5i)T + 343iT^{2} \)
11 \( 1 + (34.4 + 14.2i)T + (941. + 941. i)T^{2} \)
13 \( 1 + (-15.3 - 37.1i)T + (-1.55e3 + 1.55e3i)T^{2} \)
17 \( 1 + 103. iT - 4.91e3T^{2} \)
19 \( 1 + (12.4 + 29.9i)T + (-4.85e3 + 4.85e3i)T^{2} \)
23 \( 1 + (-72.3 + 72.3i)T - 1.21e4iT^{2} \)
29 \( 1 + (-23.9 + 9.90i)T + (1.72e4 - 1.72e4i)T^{2} \)
31 \( 1 - 124.T + 2.97e4T^{2} \)
37 \( 1 + (-18.0 + 43.5i)T + (-3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (45.1 - 45.1i)T - 6.89e4iT^{2} \)
43 \( 1 + (-457. - 189. i)T + (5.62e4 + 5.62e4i)T^{2} \)
47 \( 1 + 582. iT - 1.03e5T^{2} \)
53 \( 1 + (-395. - 163. i)T + (1.05e5 + 1.05e5i)T^{2} \)
59 \( 1 + (142. - 344. i)T + (-1.45e5 - 1.45e5i)T^{2} \)
61 \( 1 + (34.8 - 14.4i)T + (1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 + (196. - 81.4i)T + (2.12e5 - 2.12e5i)T^{2} \)
71 \( 1 + (-520. - 520. i)T + 3.57e5iT^{2} \)
73 \( 1 + (582. - 582. i)T - 3.89e5iT^{2} \)
79 \( 1 - 157. iT - 4.93e5T^{2} \)
83 \( 1 + (54.5 + 131. i)T + (-4.04e5 + 4.04e5i)T^{2} \)
89 \( 1 + (272. + 272. i)T + 7.04e5iT^{2} \)
97 \( 1 - 788.T + 9.12e5T^{2} \)
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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

−12.88761428370351040394977980831, −11.54691772335094607054075946380, −10.32486314408303697329896741862, −9.216375705617876548259616017328, −8.459187048077988215157283114351, −7.11304610973792403024703738743, −5.60745376384868623172369197679, −4.56231194661571956040882462552, −2.64535885938283581896574180904, −0.65273658823773463938822981810, 2.53450037443163660957469721745, 3.38422840719244477327964259466, 5.76042949963763294721724058886, 6.37159878568443959212228648019, 7.84260058959697824564828905492, 9.128092946138558000568048164118, 10.18910886151327367574941335255, 10.87457186572583842002939682878, 12.31597441410525911758547413820, 13.20034182269929837182761679271

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