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Properties

Label 2-7-7.6-c16-0-1
Degree $2$
Conductor $7$
Sign $-0.865 - 0.501i$
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  + 270.·2-s + 8.29e3i·3-s + 7.37e3·4-s + 2.34e5i·5-s + 2.23e6i·6-s + (−4.98e6 − 2.89e6i)7-s − 1.57e7·8-s − 2.56e7·9-s + 6.34e7i·10-s + 5.66e7·11-s + 6.11e7i·12-s + 5.49e8i·13-s + (−1.34e9 − 7.81e8i)14-s − 1.94e9·15-s − 4.72e9·16-s + 1.34e10i·17-s + ⋯
L(s)  = 1  + 1.05·2-s + 1.26i·3-s + 0.112·4-s + 0.601i·5-s + 1.33i·6-s + (−0.865 − 0.501i)7-s − 0.936·8-s − 0.596·9-s + 0.634i·10-s + 0.264·11-s + 0.142i·12-s + 0.673i·13-s + (−0.912 − 0.529i)14-s − 0.759·15-s − 1.09·16-s + 1.93i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $-0.865 - 0.501i$
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: \((2,\ 7,\ (\ :8),\ -0.865 - 0.501i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.494216 + 1.83698i\)
\(L(\frac12)\) \(\approx\) \(0.494216 + 1.83698i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (4.98e6 + 2.89e6i)T \)
good2 \( 1 - 270.T + 6.55e4T^{2} \)
3 \( 1 - 8.29e3iT - 4.30e7T^{2} \)
5 \( 1 - 2.34e5iT - 1.52e11T^{2} \)
11 \( 1 - 5.66e7T + 4.59e16T^{2} \)
13 \( 1 - 5.49e8iT - 6.65e17T^{2} \)
17 \( 1 - 1.34e10iT - 4.86e19T^{2} \)
19 \( 1 + 4.64e8iT - 2.88e20T^{2} \)
23 \( 1 - 6.50e10T + 6.13e21T^{2} \)
29 \( 1 - 8.11e10T + 2.50e23T^{2} \)
31 \( 1 + 1.17e12iT - 7.27e23T^{2} \)
37 \( 1 + 2.93e12T + 1.23e25T^{2} \)
41 \( 1 - 7.81e12iT - 6.37e25T^{2} \)
43 \( 1 - 1.54e13T + 1.36e26T^{2} \)
47 \( 1 - 1.35e13iT - 5.66e26T^{2} \)
53 \( 1 + 1.02e14T + 3.87e27T^{2} \)
59 \( 1 - 1.65e14iT - 2.15e28T^{2} \)
61 \( 1 + 3.07e14iT - 3.67e28T^{2} \)
67 \( 1 - 3.75e13T + 1.64e29T^{2} \)
71 \( 1 + 1.78e14T + 4.16e29T^{2} \)
73 \( 1 + 8.37e14iT - 6.50e29T^{2} \)
79 \( 1 - 2.06e15T + 2.30e30T^{2} \)
83 \( 1 - 2.72e15iT - 5.07e30T^{2} \)
89 \( 1 + 6.65e15iT - 1.54e31T^{2} \)
97 \( 1 + 3.48e15iT - 6.14e31T^{2} \)
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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

−19.10808000212083594102951360787, −16.93617765336862646246782471938, −15.42651180672551070318452908363, −14.42772308987498462527910367422, −12.85469148295912301339798815484, −10.79299783090624897734378993053, −9.353112575074517698645663269631, −6.30561153151956557858984688209, −4.36514625967881302143236767949, −3.36150270704629977427399245289, 0.61691672753385549122828254274, 2.90449222893751221867485214671, 5.22868071708704140629903422200, 6.84716971875752101635523234289, 9.031092644427658000195002381015, 12.13556671967853401506416125630, 12.85208652775765535123977129619, 13.95760274947659870175120743992, 15.83942403291452723319735481460, 17.87153789712232846309158920274

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