Properties

Label 2-7-7.3-c4-0-1
Degree $2$
Conductor $7$
Sign $0.425 + 0.904i$
Analytic cond. $0.723589$
Root an. cond. $0.850640$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.34 − 5.79i)2-s + (8.53 + 4.92i)3-s + (−14.3 + 24.9i)4-s + (6.57 − 3.79i)5-s − 65.9i·6-s + (−32.8 + 36.3i)7-s + 85.3·8-s + (8.07 + 13.9i)9-s + (−43.9 − 25.3i)10-s + (25.3 − 43.9i)11-s + (−245. + 141. i)12-s − 7.50i·13-s + (320. + 68.5i)14-s + 74.7·15-s + (−55.5 − 96.1i)16-s + (−300. − 173. i)17-s + ⋯
L(s)  = 1  + (−0.836 − 1.44i)2-s + (0.948 + 0.547i)3-s + (−0.898 + 1.55i)4-s + (0.262 − 0.151i)5-s − 1.83i·6-s + (−0.670 + 0.742i)7-s + 1.33·8-s + (0.0996 + 0.172i)9-s + (−0.439 − 0.253i)10-s + (0.209 − 0.363i)11-s + (−1.70 + 0.984i)12-s − 0.0444i·13-s + (1.63 + 0.349i)14-s + 0.332·15-s + (−0.216 − 0.375i)16-s + (−1.04 − 0.600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7\)
Sign: $0.425 + 0.904i$
Analytic conductor: \(0.723589\)
Root analytic conductor: \(0.850640\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{7} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7,\ (\ :2),\ 0.425 + 0.904i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.649594 - 0.412319i\)
\(L(\frac12)\) \(\approx\) \(0.649594 - 0.412319i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (32.8 - 36.3i)T \)
good2 \( 1 + (3.34 + 5.79i)T + (-8 + 13.8i)T^{2} \)
3 \( 1 + (-8.53 - 4.92i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (-6.57 + 3.79i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (-25.3 + 43.9i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + 7.50iT - 2.85e4T^{2} \)
17 \( 1 + (300. + 173. i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-420. + 242. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-168. - 292. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 215.T + 7.07e5T^{2} \)
31 \( 1 + (-1.10e3 - 640. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (-176. - 305. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 1.04e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.02e3T + 3.41e6T^{2} \)
47 \( 1 + (1.70e3 - 982. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (1.92e3 - 3.33e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-3.79e3 - 2.18e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-1.61e3 + 934. i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (223. - 387. i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 1.48e3T + 2.54e7T^{2} \)
73 \( 1 + (610. + 352. i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (2.85e3 + 4.93e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 4.02e3iT - 4.74e7T^{2} \)
89 \( 1 + (-5.85e3 + 3.37e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 1.15e4iT - 8.85e7T^{2} \)
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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

−21.20443280839632438410738218049, −20.11205165910697033633138975099, −19.18448667915759720334624006054, −17.73075558417224071025640254096, −15.60934794768144622514547500379, −13.44169416177175947844631363405, −11.64038034658922173593467619244, −9.652166542892594551482947676731, −8.868463082374881474098087804275, −3.01511646925818929048655556808, 6.69324740543004824973130781459, 8.135322344559972522236817182486, 9.778335567515717376235668356354, 13.45744470465150011139308882123, 14.65440519442219863039888674072, 16.24072330190243185278625729937, 17.61458772534810586953838633892, 18.97946947601191640290117603898, 20.12791688828009406062574847489, 22.69096249823631940536851598019

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