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Properties

Label 2-7-7.6-c8-0-2
Degree $2$
Conductor $7$
Sign $0.820 - 0.570i$
Analytic cond. $2.85165$
Root an. cond. $1.68868$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 21.5·2-s + 119. i·3-s + 209.·4-s − 786. i·5-s + 2.58e3i·6-s + (1.97e3 − 1.37e3i)7-s − 1.01e3·8-s − 7.80e3·9-s − 1.69e4i·10-s − 6.46e3·11-s + 2.50e4i·12-s + 2.33e4i·13-s + (4.25e4 − 2.95e4i)14-s + 9.43e4·15-s − 7.53e4·16-s + 8.53e4i·17-s + ⋯
L(s)  = 1  + 1.34·2-s + 1.47i·3-s + 0.816·4-s − 1.25i·5-s + 1.99i·6-s + (0.820 − 0.570i)7-s − 0.247·8-s − 1.19·9-s − 1.69i·10-s − 0.441·11-s + 1.20i·12-s + 0.815i·13-s + (1.10 − 0.769i)14-s + 1.86·15-s − 1.14·16-s + 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.570i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.820 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $0.820 - 0.570i$
Analytic conductor: \(2.85165\)
Root analytic conductor: \(1.68868\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :4),\ 0.820 - 0.570i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.31229 + 0.724986i\)
\(L(\frac12)\) \(\approx\) \(2.31229 + 0.724986i\)
\(L(5)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.97e3 + 1.37e3i)T \)
good2 \( 1 - 21.5T + 256T^{2} \)
3 \( 1 - 119. iT - 6.56e3T^{2} \)
5 \( 1 + 786. iT - 3.90e5T^{2} \)
11 \( 1 + 6.46e3T + 2.14e8T^{2} \)
13 \( 1 - 2.33e4iT - 8.15e8T^{2} \)
17 \( 1 - 8.53e4iT - 6.97e9T^{2} \)
19 \( 1 + 1.71e5iT - 1.69e10T^{2} \)
23 \( 1 - 1.45e5T + 7.83e10T^{2} \)
29 \( 1 + 6.84e5T + 5.00e11T^{2} \)
31 \( 1 - 5.97e5iT - 8.52e11T^{2} \)
37 \( 1 - 2.08e6T + 3.51e12T^{2} \)
41 \( 1 + 2.26e5iT - 7.98e12T^{2} \)
43 \( 1 - 3.79e6T + 1.16e13T^{2} \)
47 \( 1 - 1.05e5iT - 2.38e13T^{2} \)
53 \( 1 + 3.46e6T + 6.22e13T^{2} \)
59 \( 1 + 1.35e7iT - 1.46e14T^{2} \)
61 \( 1 - 8.42e5iT - 1.91e14T^{2} \)
67 \( 1 - 9.62e6T + 4.06e14T^{2} \)
71 \( 1 + 4.99e6T + 6.45e14T^{2} \)
73 \( 1 - 2.22e7iT - 8.06e14T^{2} \)
79 \( 1 + 3.48e7T + 1.51e15T^{2} \)
83 \( 1 + 8.16e7iT - 2.25e15T^{2} \)
89 \( 1 - 7.05e7iT - 3.93e15T^{2} \)
97 \( 1 + 3.15e7iT - 7.83e15T^{2} \)
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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

−21.12201806840665937203406467935, −20.28143750916331967545768382331, −17.11690275853842320758844558402, −15.82883695704377801572198023599, −14.58713763233924537093935352423, −13.03208030200048460915389861169, −11.16239162688341850075024894652, −9.030410947693021566905195242667, −5.07748792087393918729663418095, −4.20506806155775639034222981459, 2.62196558151519084848169734432, 5.84779613018174459984437509586, 7.54794770455962958142649190866, 11.39779617772281970880187381150, 12.71875763028566089329671539017, 14.04788547115477792732090681071, 15.04227363600873394567184645294, 18.08553071687263304072017114947, 18.64122357256180398537132390360, 20.77709251479315866811652764166

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