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Properties

Label 2-7-7.5-c6-0-2
Degree $2$
Conductor $7$
Sign $0.386 + 0.922i$
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  + (6 − 10.3i)2-s + (−10.5 + 6.06i)3-s + (−40 − 69.2i)4-s + (157.5 + 90.9i)5-s + 145. i·6-s − 343·7-s − 192.·8-s + (−291 + 504. i)9-s + (1.89e3 − 1.09e3i)10-s + (−739.5 − 1.28e3i)11-s + (840 + 484. i)12-s − 484. i·13-s + (−2.05e3 + 3.56e3i)14-s − 2.20e3·15-s + (1.40e3 − 2.43e3i)16-s + (−2.61e3 + 1.50e3i)17-s + ⋯
L(s)  = 1  + (0.750 − 1.29i)2-s + (−0.388 + 0.224i)3-s + (−0.625 − 1.08i)4-s + (1.26 + 0.727i)5-s + 0.673i·6-s − 7-s − 0.375·8-s + (−0.399 + 0.691i)9-s + (1.89 − 1.09i)10-s + (−0.555 − 0.962i)11-s + (0.486 + 0.280i)12-s − 0.220i·13-s + (−0.750 + 1.29i)14-s − 0.653·15-s + (0.343 − 0.595i)16-s + (−0.532 + 0.307i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(1.61037\)
Root analytic conductor: \(1.26900\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :3),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.28443 - 0.854380i\)
\(L(\frac12)\) \(\approx\) \(1.28443 - 0.854380i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + 343T \)
good2 \( 1 + (-6 + 10.3i)T + (-32 - 55.4i)T^{2} \)
3 \( 1 + (10.5 - 6.06i)T + (364.5 - 631. i)T^{2} \)
5 \( 1 + (-157.5 - 90.9i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (739.5 + 1.28e3i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 484. iT - 4.82e6T^{2} \)
17 \( 1 + (2.61e3 - 1.50e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-5.95e3 - 3.43e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-2.95e3 + 5.12e3i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 3.97e3T + 5.94e8T^{2} \)
31 \( 1 + (1.10e4 - 6.40e3i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-3.07e4 + 5.33e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 1.10e5iT - 4.75e9T^{2} \)
43 \( 1 + 1.74e4T + 6.32e9T^{2} \)
47 \( 1 + (2.65e4 + 1.53e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (-3.02e4 - 5.24e4i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (1.86e5 - 1.07e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-1.40e5 - 8.13e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-1.34e5 - 2.32e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 1.01e5T + 1.28e11T^{2} \)
73 \( 1 + (-2.75e5 + 1.58e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (1.81e5 - 3.13e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + 2.16e5iT - 3.26e11T^{2} \)
89 \( 1 + (1.15e6 + 6.67e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 1.51e6iT - 8.32e11T^{2} \)
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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

−21.42200153598462369562707372982, −19.76607650522534247978860423039, −18.41341380025591701176628862167, −16.50111391846880863031818652137, −14.02163605348064336224635231819, −13.06725590923923553274413397812, −11.03921504661519098143836210849, −10.02749283441153501165187102793, −5.74983451610027886989667333927, −2.79702464582799503960219017120, 5.35236797248870929593580240766, 6.73625150965515819440507321704, 9.457427395277246449607015568183, 12.68512126154162032588112741914, 13.70429933300513078472115009033, 15.49348055359627014559570455228, 16.86224901203694508411138291154, 17.84228833585390779634555733243, 20.38002111999181750100963248619, 22.05933765386141182914082004724

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