Properties

Label 2-7-7.2-c9-0-0
Degree $2$
Conductor $7$
Sign $-0.989 + 0.141i$
Analytic cond. $3.60525$
Root an. cond. $1.89874$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−19.4 + 33.6i)2-s + (113. + 196. i)3-s + (−499. − 865. i)4-s + (−162. + 281. i)5-s − 8.84e3·6-s + (5.23e3 − 3.59e3i)7-s + 1.89e4·8-s + (−1.60e4 + 2.77e4i)9-s + (−6.32e3 − 1.09e4i)10-s + (2.44e4 + 4.24e4i)11-s + (1.13e5 − 1.96e5i)12-s − 1.08e5·13-s + (1.93e4 + 2.46e5i)14-s − 7.40e4·15-s + (−1.12e5 + 1.94e5i)16-s + (1.14e5 + 1.97e5i)17-s + ⋯
L(s)  = 1  + (−0.859 + 1.48i)2-s + (0.810 + 1.40i)3-s + (−0.975 − 1.69i)4-s + (−0.116 + 0.201i)5-s − 2.78·6-s + (0.824 − 0.566i)7-s + 1.63·8-s + (−0.814 + 1.41i)9-s + (−0.200 − 0.346i)10-s + (0.504 + 0.873i)11-s + (1.58 − 2.74i)12-s − 1.05·13-s + (0.134 + 1.71i)14-s − 0.377·15-s + (−0.429 + 0.743i)16-s + (0.331 + 0.574i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.141i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.141i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $-0.989 + 0.141i$
Analytic conductor: \(3.60525\)
Root analytic conductor: \(1.89874\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :9/2),\ -0.989 + 0.141i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.0794313 - 1.11694i\)
\(L(\frac12)\) \(\approx\) \(0.0794313 - 1.11694i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-5.23e3 + 3.59e3i)T \)
good2 \( 1 + (19.4 - 33.6i)T + (-256 - 443. i)T^{2} \)
3 \( 1 + (-113. - 196. i)T + (-9.84e3 + 1.70e4i)T^{2} \)
5 \( 1 + (162. - 281. i)T + (-9.76e5 - 1.69e6i)T^{2} \)
11 \( 1 + (-2.44e4 - 4.24e4i)T + (-1.17e9 + 2.04e9i)T^{2} \)
13 \( 1 + 1.08e5T + 1.06e10T^{2} \)
17 \( 1 + (-1.14e5 - 1.97e5i)T + (-5.92e10 + 1.02e11i)T^{2} \)
19 \( 1 + (-2.84e4 + 4.92e4i)T + (-1.61e11 - 2.79e11i)T^{2} \)
23 \( 1 + (-7.21e5 + 1.25e6i)T + (-9.00e11 - 1.55e12i)T^{2} \)
29 \( 1 + 2.66e5T + 1.45e13T^{2} \)
31 \( 1 + (8.07e4 + 1.39e5i)T + (-1.32e13 + 2.28e13i)T^{2} \)
37 \( 1 + (-3.04e6 + 5.27e6i)T + (-6.49e13 - 1.12e14i)T^{2} \)
41 \( 1 - 1.73e7T + 3.27e14T^{2} \)
43 \( 1 - 1.34e7T + 5.02e14T^{2} \)
47 \( 1 + (9.59e5 - 1.66e6i)T + (-5.59e14 - 9.69e14i)T^{2} \)
53 \( 1 + (5.02e7 + 8.70e7i)T + (-1.64e15 + 2.85e15i)T^{2} \)
59 \( 1 + (-7.83e7 - 1.35e8i)T + (-4.33e15 + 7.50e15i)T^{2} \)
61 \( 1 + (3.73e7 - 6.47e7i)T + (-5.84e15 - 1.01e16i)T^{2} \)
67 \( 1 + (6.10e7 + 1.05e8i)T + (-1.36e16 + 2.35e16i)T^{2} \)
71 \( 1 + 1.47e8T + 4.58e16T^{2} \)
73 \( 1 + (1.78e8 + 3.08e8i)T + (-2.94e16 + 5.09e16i)T^{2} \)
79 \( 1 + (-2.81e8 + 4.87e8i)T + (-5.99e16 - 1.03e17i)T^{2} \)
83 \( 1 + 2.10e8T + 1.86e17T^{2} \)
89 \( 1 + (-2.74e8 + 4.74e8i)T + (-1.75e17 - 3.03e17i)T^{2} \)
97 \( 1 - 9.75e8T + 7.60e17T^{2} \)
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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

−20.90291337788736844607270367539, −19.49252000558934333655514656059, −17.51891415734595628591471629016, −16.46340546179272124885178981078, −14.82294086720128682187048193307, −14.65695498896229440423177759393, −10.33349111562686108345758050704, −9.062504312230013678758404662241, −7.50562393348574146132687568386, −4.68124674324458104047962861095, 1.10702421639492505080252350268, 2.65236173140569496900342592272, 7.87453610745396513767716940570, 9.102199684904835456817141272323, 11.54613287655746489648624342940, 12.58201324352729634435125898386, 14.20284318510304051555282182089, 17.43193042016528496123474811177, 18.60160918982480101579975229751, 19.37870369569491336130151125871

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