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Properties

Label 2-7-7.6-c16-0-2
Degree $2$
Conductor $7$
Sign $0.152 - 0.988i$
Analytic cond. $11.3627$
Root an. cond. $3.37086$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 408.·2-s + 7.47e3i·3-s + 1.01e5·4-s − 2.58e4i·5-s − 3.05e6i·6-s + (8.79e5 − 5.69e6i)7-s − 1.45e7·8-s − 1.28e7·9-s + 1.05e7i·10-s + 2.58e8·11-s + 7.56e8i·12-s + 7.88e8i·13-s + (−3.58e8 + 2.32e9i)14-s + 1.93e8·15-s − 6.89e8·16-s − 8.43e9i·17-s + ⋯
L(s)  = 1  − 1.59·2-s + 1.13i·3-s + 1.54·4-s − 0.0662i·5-s − 1.81i·6-s + (0.152 − 0.988i)7-s − 0.867·8-s − 0.299·9-s + 0.105i·10-s + 1.20·11-s + 1.75i·12-s + 0.966i·13-s + (−0.243 + 1.57i)14-s + 0.0755·15-s − 0.160·16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.152 - 0.988i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $0.152 - 0.988i$
Analytic conductor: \(11.3627\)
Root analytic conductor: \(3.37086\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :8),\ 0.152 - 0.988i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.628568 + 0.539005i\)
\(L(\frac12)\) \(\approx\) \(0.628568 + 0.539005i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-8.79e5 + 5.69e6i)T \)
good2 \( 1 + 408.T + 6.55e4T^{2} \)
3 \( 1 - 7.47e3iT - 4.30e7T^{2} \)
5 \( 1 + 2.58e4iT - 1.52e11T^{2} \)
11 \( 1 - 2.58e8T + 4.59e16T^{2} \)
13 \( 1 - 7.88e8iT - 6.65e17T^{2} \)
17 \( 1 + 8.43e9iT - 4.86e19T^{2} \)
19 \( 1 - 2.52e10iT - 2.88e20T^{2} \)
23 \( 1 + 9.65e9T + 6.13e21T^{2} \)
29 \( 1 + 6.70e11T + 2.50e23T^{2} \)
31 \( 1 - 7.88e11iT - 7.27e23T^{2} \)
37 \( 1 - 4.88e12T + 1.23e25T^{2} \)
41 \( 1 + 2.16e11iT - 6.37e25T^{2} \)
43 \( 1 - 5.58e12T + 1.36e26T^{2} \)
47 \( 1 - 4.97e11iT - 5.66e26T^{2} \)
53 \( 1 - 4.55e13T + 3.87e27T^{2} \)
59 \( 1 - 1.09e14iT - 2.15e28T^{2} \)
61 \( 1 - 3.61e14iT - 3.67e28T^{2} \)
67 \( 1 - 2.27e14T + 1.64e29T^{2} \)
71 \( 1 - 6.95e14T + 4.16e29T^{2} \)
73 \( 1 - 1.27e15iT - 6.50e29T^{2} \)
79 \( 1 + 1.01e14T + 2.30e30T^{2} \)
83 \( 1 + 9.27e14iT - 5.07e30T^{2} \)
89 \( 1 + 2.66e15iT - 1.54e31T^{2} \)
97 \( 1 + 6.68e15iT - 6.14e31T^{2} \)
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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

−18.49616019196979486459495750097, −16.77408442660099007228122701016, −16.38856390406818443635391531846, −14.39739324652389401865079888116, −11.34632020390951750262436360325, −10.04287750044170554673410681151, −9.069254155269350530451503741832, −7.14312278983701715854460046081, −4.13767570441559325401474673432, −1.22499823976818890150986452249, 0.76780744705105217772819756926, 2.11287417488916144416667247635, 6.45730162879028909105975705868, 7.938054754982855723216317295690, 9.242876164183163286650148421017, 11.24682006233854260842764702504, 12.81167990489659843060735272396, 15.17249571298602646997319431013, 17.09209932844076510523462330164, 18.08463344727583969232744998572

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