Properties

Label 2-7-7.4-c9-0-3
Degree $2$
Conductor $7$
Sign $-0.000219 + 0.999i$
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  + (−2.74 − 4.75i)2-s + (−1.70 + 2.94i)3-s + (240. − 417. i)4-s + (−828. − 1.43e3i)5-s + 18.6·6-s + (2.82e3 − 5.69e3i)7-s − 5.45e3·8-s + (9.83e3 + 1.70e4i)9-s + (−4.54e3 + 7.87e3i)10-s + (−1.68e4 + 2.91e4i)11-s + (820. + 1.42e3i)12-s + 4.79e4·13-s + (−3.47e4 + 2.20e3i)14-s + 5.64e3·15-s + (−1.08e5 − 1.87e5i)16-s + (1.72e5 − 2.99e5i)17-s + ⋯
L(s)  = 1  + (−0.121 − 0.210i)2-s + (−0.0121 + 0.0210i)3-s + (0.470 − 0.815i)4-s + (−0.593 − 1.02i)5-s + 0.00588·6-s + (0.444 − 0.895i)7-s − 0.470·8-s + (0.499 + 0.865i)9-s + (−0.143 + 0.249i)10-s + (−0.346 + 0.600i)11-s + (0.0114 + 0.0197i)12-s + 0.465·13-s + (−0.242 + 0.0153i)14-s + 0.0288·15-s + (−0.413 − 0.716i)16-s + (0.502 − 0.869i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.000219 + 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.000219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $-0.000219 + 0.999i$
Analytic conductor: \(3.60525\)
Root analytic conductor: \(1.89874\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :9/2),\ -0.000219 + 0.999i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.962071 - 0.962282i\)
\(L(\frac12)\) \(\approx\) \(0.962071 - 0.962282i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.82e3 + 5.69e3i)T \)
good2 \( 1 + (2.74 + 4.75i)T + (-256 + 443. i)T^{2} \)
3 \( 1 + (1.70 - 2.94i)T + (-9.84e3 - 1.70e4i)T^{2} \)
5 \( 1 + (828. + 1.43e3i)T + (-9.76e5 + 1.69e6i)T^{2} \)
11 \( 1 + (1.68e4 - 2.91e4i)T + (-1.17e9 - 2.04e9i)T^{2} \)
13 \( 1 - 4.79e4T + 1.06e10T^{2} \)
17 \( 1 + (-1.72e5 + 2.99e5i)T + (-5.92e10 - 1.02e11i)T^{2} \)
19 \( 1 + (-2.02e5 - 3.50e5i)T + (-1.61e11 + 2.79e11i)T^{2} \)
23 \( 1 + (-9.26e5 - 1.60e6i)T + (-9.00e11 + 1.55e12i)T^{2} \)
29 \( 1 - 6.82e5T + 1.45e13T^{2} \)
31 \( 1 + (-4.54e6 + 7.87e6i)T + (-1.32e13 - 2.28e13i)T^{2} \)
37 \( 1 + (-7.77e6 - 1.34e7i)T + (-6.49e13 + 1.12e14i)T^{2} \)
41 \( 1 + 2.98e7T + 3.27e14T^{2} \)
43 \( 1 - 6.28e6T + 5.02e14T^{2} \)
47 \( 1 + (5.16e6 + 8.93e6i)T + (-5.59e14 + 9.69e14i)T^{2} \)
53 \( 1 + (3.32e7 - 5.75e7i)T + (-1.64e15 - 2.85e15i)T^{2} \)
59 \( 1 + (3.52e7 - 6.11e7i)T + (-4.33e15 - 7.50e15i)T^{2} \)
61 \( 1 + (-4.21e7 - 7.30e7i)T + (-5.84e15 + 1.01e16i)T^{2} \)
67 \( 1 + (-1.05e8 + 1.83e8i)T + (-1.36e16 - 2.35e16i)T^{2} \)
71 \( 1 - 2.31e7T + 4.58e16T^{2} \)
73 \( 1 + (1.24e8 - 2.15e8i)T + (-2.94e16 - 5.09e16i)T^{2} \)
79 \( 1 + (1.33e8 + 2.30e8i)T + (-5.99e16 + 1.03e17i)T^{2} \)
83 \( 1 - 6.33e8T + 1.86e17T^{2} \)
89 \( 1 + (3.11e8 + 5.38e8i)T + (-1.75e17 + 3.03e17i)T^{2} \)
97 \( 1 + 9.94e8T + 7.60e17T^{2} \)
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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

−20.11453679213555019878465768203, −18.72995575333795588436883095465, −16.70275458677987113424494062811, −15.48662234571430652597936980886, −13.55296672243535620254730089741, −11.59799708554619691838692389789, −10.00950153722469222705011779844, −7.66272054885647040106170760717, −4.86994364224670782644975698177, −1.18089554118922301207582774195, 3.17932847120696140692856041184, 6.66734352418330640950634354107, 8.410426124784862540419246889968, 11.08396849916735373559901030186, 12.42607014030712190759746878495, 14.87531363141998757822036095919, 15.89913772913792454212114134572, 17.83018431176297885882959290598, 18.90100800902004101758284723359, 20.93136809069602134418584109765

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