Properties

Label 2-7-7.6-c6-0-0
Degree $2$
Conductor $7$
Sign $-0.387 - 0.921i$
Analytic cond. $1.61037$
Root an. cond. $1.26900$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 45.1i·3-s − 45.1i·5-s − 361. i·6-s + (133 + 316. i)7-s + 512·8-s − 1.31e3·9-s + 361. i·10-s + 874·11-s + 2.21e3i·13-s + (−1.06e3 − 2.52e3i)14-s + 2.03e3·15-s − 4.09e3·16-s − 5.96e3i·17-s + 1.04e4·18-s + 3.11e3i·19-s + ⋯
L(s)  = 1  − 2-s + 1.67i·3-s − 0.361i·5-s − 1.67i·6-s + (0.387 + 0.921i)7-s + 8-s − 1.79·9-s + 0.361i·10-s + 0.656·11-s + 1.00i·13-s + (−0.387 − 0.921i)14-s + 0.604·15-s − 16-s − 1.21i·17-s + 1.79·18-s + 0.454i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $-0.387 - 0.921i$
Analytic conductor: \(1.61037\)
Root analytic conductor: \(1.26900\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :3),\ -0.387 - 0.921i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.369522 + 0.556333i\)
\(L(\frac12)\) \(\approx\) \(0.369522 + 0.556333i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-133 - 316. i)T \)
good2 \( 1 + 8T + 64T^{2} \)
3 \( 1 - 45.1iT - 729T^{2} \)
5 \( 1 + 45.1iT - 1.56e4T^{2} \)
11 \( 1 - 874T + 1.77e6T^{2} \)
13 \( 1 - 2.21e3iT - 4.82e6T^{2} \)
17 \( 1 + 5.96e3iT - 2.41e7T^{2} \)
19 \( 1 - 3.11e3iT - 4.70e7T^{2} \)
23 \( 1 - 4.73e3T + 1.48e8T^{2} \)
29 \( 1 - 1.11e4T + 5.94e8T^{2} \)
31 \( 1 + 2.74e4iT - 8.87e8T^{2} \)
37 \( 1 - 3.00e3T + 2.56e9T^{2} \)
41 \( 1 - 5.75e4iT - 4.75e9T^{2} \)
43 \( 1 - 3.14e4T + 6.32e9T^{2} \)
47 \( 1 + 7.24e4iT - 1.07e10T^{2} \)
53 \( 1 + 7.64e4T + 2.21e10T^{2} \)
59 \( 1 - 1.13e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.75e5iT - 5.15e10T^{2} \)
67 \( 1 - 4.95e5T + 9.04e10T^{2} \)
71 \( 1 + 1.84e5T + 1.28e11T^{2} \)
73 \( 1 + 6.09e4iT - 1.51e11T^{2} \)
79 \( 1 + 5.34e5T + 2.43e11T^{2} \)
83 \( 1 + 7.14e5iT - 3.26e11T^{2} \)
89 \( 1 + 6.29e5iT - 4.96e11T^{2} \)
97 \( 1 - 8.14e5iT - 8.32e11T^{2} \)
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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

−21.56593118546249998153867639690, −20.33687755471626900603484997699, −18.67158597206423904655628524524, −17.00653500585425297663577113249, −16.03839533185168513375849669959, −14.42419963223029106500639446527, −11.40316797027950834772575422884, −9.625020434465131307411498781647, −8.798957585047171515468592210723, −4.71547017025821628032419045203, 1.07733282911391212732378645122, 7.01191900691460940820953897559, 8.311442177711380676451943091851, 10.75923984241788276839323529391, 12.86581600670512219937921367609, 14.20134740872490013054366128409, 17.19068626466928186347118288612, 17.82701406890360998772407454060, 19.11798010630834252916645370873, 19.97041013568860581398130146807

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