↑

Properties

Label 2-154-7.4-c3-0-14
Degree $2$
Conductor $154$
Sign $-0.523 + 0.851i$
Analytic cond. $9.08629$
Root an. cond. $3.01434$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − 1.73i)2-s + (−0.644 + 1.11i)3-s + (−1.99 + 3.46i)4-s + (−3.59 − 6.22i)5-s + 2.57·6-s + (15.6 − 9.83i)7-s + 7.99·8-s + (12.6 + 21.9i)9-s + (−7.19 + 12.4i)10-s + (5.5 − 9.52i)11-s + (−2.57 − 4.46i)12-s − 53.9·13-s + (−32.7 − 17.3i)14-s + 9.26·15-s + (−8 − 13.8i)16-s + (48.8 − 84.6i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.124 + 0.214i)3-s + (−0.249 + 0.433i)4-s + (−0.321 − 0.557i)5-s + 0.175·6-s + (0.847 − 0.530i)7-s + 0.353·8-s + (0.469 + 0.812i)9-s + (−0.227 + 0.393i)10-s + (0.150 − 0.261i)11-s + (−0.0620 − 0.107i)12-s − 1.15·13-s + (−0.624 − 0.331i)14-s + 0.159·15-s + (−0.125 − 0.216i)16-s + (0.697 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.523 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-0.523 + 0.851i$
Analytic conductor: \(9.08629\)
Root analytic conductor: \(3.01434\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :3/2),\ -0.523 + 0.851i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.522846 - 0.934953i\)
\(L(\frac12)\) \(\approx\) \(0.522846 - 0.934953i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + 1.73i)T \)
7 \( 1 + (-15.6 + 9.83i)T \)
11 \( 1 + (-5.5 + 9.52i)T \)
good3 \( 1 + (0.644 - 1.11i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (3.59 + 6.22i)T + (-62.5 + 108. i)T^{2} \)
13 \( 1 + 53.9T + 2.19e3T^{2} \)
17 \( 1 + (-48.8 + 84.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (75.8 + 131. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (13.6 + 23.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 111.T + 2.43e4T^{2} \)
31 \( 1 + (-101. + 175. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (14.9 + 25.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 88.4T + 6.89e4T^{2} \)
43 \( 1 + 116.T + 7.95e4T^{2} \)
47 \( 1 + (-272. - 471. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-119. + 206. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-10.8 + 18.8i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-121. - 210. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (499. - 865. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 910.T + 3.57e5T^{2} \)
73 \( 1 + (-530. + 918. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (461. + 800. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 588.T + 5.71e5T^{2} \)
89 \( 1 + (-125. - 217. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 255.T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.93770124218240666946401084621, −11.15764606687531632580583095293, −10.23405923095476586634452745149, −9.187452150982529249911121057841, −8.015389998695131314920043760691, −7.20873759310002444491947325651, −4.99417083840847829997872371511, −4.38378209067299033277020973679, −2.38799234878783124370385846248, −0.60729670775369520171745487498, 1.69815775845014620829893396701, 3.86100925634247863442681146368, 5.38932249671823563773945787853, 6.56317656038961433816019566217, 7.57849084937570845198950040431, 8.472049724902948240557162713527, 9.770665302254438142335575877382, 10.65377616035812950961821202174, 12.02225224591818137887072049894, 12.55764763991378640044959903777

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