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Properties

Label 2-7-7.6-c16-0-5
Degree $2$
Conductor $7$
Sign $0.903 + 0.429i$
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  + 40.3·2-s + 6.14e3i·3-s − 6.39e4·4-s − 6.23e5i·5-s + 2.47e5i·6-s + (5.20e6 + 2.47e6i)7-s − 5.21e6·8-s + 5.25e6·9-s − 2.51e7i·10-s + 2.60e8·11-s − 3.92e8i·12-s − 3.22e7i·13-s + (2.09e8 + 9.97e7i)14-s + 3.83e9·15-s + 3.97e9·16-s − 4.55e9i·17-s + ⋯
L(s)  = 1  + 0.157·2-s + 0.936i·3-s − 0.975·4-s − 1.59i·5-s + 0.147i·6-s + (0.903 + 0.429i)7-s − 0.311·8-s + 0.122·9-s − 0.251i·10-s + 1.21·11-s − 0.913i·12-s − 0.0395i·13-s + (0.142 + 0.0675i)14-s + 1.49·15-s + 0.926·16-s − 0.652i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $0.903 + 0.429i$
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: \((2,\ 7,\ (\ :8),\ 0.903 + 0.429i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(1.68351 - 0.379522i\)
\(L(\frac12)\) \(\approx\) \(1.68351 - 0.379522i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-5.20e6 - 2.47e6i)T \)
good2 \( 1 - 40.3T + 6.55e4T^{2} \)
3 \( 1 - 6.14e3iT - 4.30e7T^{2} \)
5 \( 1 + 6.23e5iT - 1.52e11T^{2} \)
11 \( 1 - 2.60e8T + 4.59e16T^{2} \)
13 \( 1 + 3.22e7iT - 6.65e17T^{2} \)
17 \( 1 + 4.55e9iT - 4.86e19T^{2} \)
19 \( 1 + 3.21e10iT - 2.88e20T^{2} \)
23 \( 1 + 2.53e10T + 6.13e21T^{2} \)
29 \( 1 - 6.07e11T + 2.50e23T^{2} \)
31 \( 1 + 6.88e11iT - 7.27e23T^{2} \)
37 \( 1 - 1.13e12T + 1.23e25T^{2} \)
41 \( 1 - 7.07e12iT - 6.37e25T^{2} \)
43 \( 1 + 1.00e13T + 1.36e26T^{2} \)
47 \( 1 + 3.10e13iT - 5.66e26T^{2} \)
53 \( 1 - 2.29e13T + 3.87e27T^{2} \)
59 \( 1 - 1.29e14iT - 2.15e28T^{2} \)
61 \( 1 - 8.46e13iT - 3.67e28T^{2} \)
67 \( 1 + 9.70e13T + 1.64e29T^{2} \)
71 \( 1 + 2.60e14T + 4.16e29T^{2} \)
73 \( 1 + 2.61e14iT - 6.50e29T^{2} \)
79 \( 1 - 2.57e15T + 2.30e30T^{2} \)
83 \( 1 + 4.41e14iT - 5.07e30T^{2} \)
89 \( 1 - 4.20e15iT - 1.54e31T^{2} \)
97 \( 1 + 9.38e15iT - 6.14e31T^{2} \)
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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

−17.83812565963491573907049569676, −16.64350062276178311534623792494, −15.12242784516554106773276628317, −13.46831978964433525795835074977, −11.87621421129670598364549335174, −9.457501743848240557279438219084, −8.658375193304882849835170938361, −5.02030855635654651648855415242, −4.34793752280574964299309615371, −0.922821147089828579800811407946, 1.47242262100068611804344306898, 3.85572757482294050877714572877, 6.43690377662169008444479819052, 7.957162796553649366475451392771, 10.30424272487658350438844299630, 12.13051893778335638707334318204, 14.00388239132152292638765349989, 14.56150716041852573047678358585, 17.45687964350506357789134406964, 18.36507302207145897832313349422

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