Properties

Label 2-7-7.4-c11-0-2
Degree $2$
Conductor $7$
Sign $0.926 - 0.375i$
Analytic cond. $5.37840$
Root an. cond. $2.31913$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.63 + 9.76i)2-s + (−23.0 + 39.9i)3-s + (960. − 1.66e3i)4-s + (4.30e3 + 7.45e3i)5-s − 520.·6-s + (4.43e4 + 3.35e3i)7-s + 4.47e4·8-s + (8.75e4 + 1.51e5i)9-s + (−4.85e4 + 8.40e4i)10-s + (3.16e5 − 5.48e5i)11-s + (4.43e4 + 7.68e4i)12-s − 2.31e6·13-s + (2.17e5 + 4.52e5i)14-s − 3.97e5·15-s + (−1.71e6 − 2.96e6i)16-s + (5.41e5 − 9.37e5i)17-s + ⋯
L(s)  = 1  + (0.124 + 0.215i)2-s + (−0.0548 + 0.0950i)3-s + (0.468 − 0.812i)4-s + (0.616 + 1.06i)5-s − 0.0273·6-s + (0.997 + 0.0754i)7-s + 0.482·8-s + (0.493 + 0.855i)9-s + (−0.153 + 0.265i)10-s + (0.593 − 1.02i)11-s + (0.0514 + 0.0891i)12-s − 1.72·13-s + (0.107 + 0.224i)14-s − 0.135·15-s + (−0.408 − 0.708i)16-s + (0.0924 − 0.160i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $0.926 - 0.375i$
Analytic conductor: \(5.37840\)
Root analytic conductor: \(2.31913\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :11/2),\ 0.926 - 0.375i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.04208 + 0.397722i\)
\(L(\frac12)\) \(\approx\) \(2.04208 + 0.397722i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-4.43e4 - 3.35e3i)T \)
good2 \( 1 + (-5.63 - 9.76i)T + (-1.02e3 + 1.77e3i)T^{2} \)
3 \( 1 + (23.0 - 39.9i)T + (-8.85e4 - 1.53e5i)T^{2} \)
5 \( 1 + (-4.30e3 - 7.45e3i)T + (-2.44e7 + 4.22e7i)T^{2} \)
11 \( 1 + (-3.16e5 + 5.48e5i)T + (-1.42e11 - 2.47e11i)T^{2} \)
13 \( 1 + 2.31e6T + 1.79e12T^{2} \)
17 \( 1 + (-5.41e5 + 9.37e5i)T + (-1.71e13 - 2.96e13i)T^{2} \)
19 \( 1 + (-3.46e6 - 6.00e6i)T + (-5.82e13 + 1.00e14i)T^{2} \)
23 \( 1 + (-3.30e4 - 5.72e4i)T + (-4.76e14 + 8.25e14i)T^{2} \)
29 \( 1 + 1.55e8T + 1.22e16T^{2} \)
31 \( 1 + (4.72e7 - 8.19e7i)T + (-1.27e16 - 2.20e16i)T^{2} \)
37 \( 1 + (1.78e8 + 3.08e8i)T + (-8.89e16 + 1.54e17i)T^{2} \)
41 \( 1 - 7.58e8T + 5.50e17T^{2} \)
43 \( 1 + 1.10e9T + 9.29e17T^{2} \)
47 \( 1 + (1.12e9 + 1.95e9i)T + (-1.23e18 + 2.14e18i)T^{2} \)
53 \( 1 + (-1.36e9 + 2.37e9i)T + (-4.63e18 - 8.02e18i)T^{2} \)
59 \( 1 + (1.20e9 - 2.09e9i)T + (-1.50e19 - 2.61e19i)T^{2} \)
61 \( 1 + (-1.13e9 - 1.96e9i)T + (-2.17e19 + 3.76e19i)T^{2} \)
67 \( 1 + (-2.44e9 + 4.24e9i)T + (-6.10e19 - 1.05e20i)T^{2} \)
71 \( 1 + 2.89e9T + 2.31e20T^{2} \)
73 \( 1 + (8.54e9 - 1.47e10i)T + (-1.56e20 - 2.71e20i)T^{2} \)
79 \( 1 + (-1.73e10 - 3.00e10i)T + (-3.73e20 + 6.47e20i)T^{2} \)
83 \( 1 + 1.98e10T + 1.28e21T^{2} \)
89 \( 1 + (4.82e9 + 8.36e9i)T + (-1.38e21 + 2.40e21i)T^{2} \)
97 \( 1 + 3.82e10T + 7.15e21T^{2} \)
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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

−19.57516387791713616995718056289, −18.42299195071583613285391573263, −16.67673186283671176363872739963, −14.80111029194693929175095660075, −14.07143429108084764903622503693, −11.28832105962944430106284336311, −10.09150017348063025417608120652, −7.24300167608646970579028984302, −5.38012711726465898080749432040, −2.02185014701758282718215617788, 1.73879828129916646730713821894, 4.62347783636021869660208259967, 7.37950912194076272177803292455, 9.430565810516140592390579958168, 11.87622000315109728755264622091, 12.84038588626784053755971407415, 14.92480427692519731139451959354, 16.96518183786439357116560088469, 17.61383742484158853437906296107, 20.14351593454708675721478755588

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