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Properties

Label 2-7e2-7.3-c4-0-7
Degree $2$
Conductor $49$
Sign $0.895 - 0.444i$
Analytic cond. $5.06512$
Root an. cond. $2.25058$
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  + (1.34 + 2.32i)2-s + (5.53 + 3.19i)3-s + (4.38 − 7.58i)4-s + (21.5 − 12.4i)5-s + 17.1i·6-s + 66.6·8-s + (−20.0 − 34.7i)9-s + (58.0 + 33.5i)10-s + (−54.3 + 94.1i)11-s + (48.5 − 28.0i)12-s + 234. i·13-s + 159.·15-s + (19.5 + 33.8i)16-s + (−177. − 102. i)17-s + (53.9 − 93.5i)18-s + (99.8 − 57.6i)19-s + ⋯
L(s)  = 1  + (0.336 + 0.582i)2-s + (0.615 + 0.355i)3-s + (0.273 − 0.474i)4-s + (0.862 − 0.498i)5-s + 0.477i·6-s + 1.04·8-s + (−0.247 − 0.429i)9-s + (0.580 + 0.335i)10-s + (−0.449 + 0.778i)11-s + (0.336 − 0.194i)12-s + 1.39i·13-s + 0.707·15-s + (0.0762 + 0.132i)16-s + (−0.614 − 0.355i)17-s + (0.166 − 0.288i)18-s + (0.276 − 0.159i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(5.06512\)
Root analytic conductor: \(2.25058\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :2),\ 0.895 - 0.444i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.44755 + 0.573366i\)
\(L(\frac12)\) \(\approx\) \(2.44755 + 0.573366i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-1.34 - 2.32i)T + (-8 + 13.8i)T^{2} \)
3 \( 1 + (-5.53 - 3.19i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (-21.5 + 12.4i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (54.3 - 94.1i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 - 234. iT - 2.85e4T^{2} \)
17 \( 1 + (177. + 102. i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-99.8 + 57.6i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (23.6 + 40.9i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 872.T + 7.07e5T^{2} \)
31 \( 1 + (700. + 404. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (311. + 539. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 1.72e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.25e3T + 3.41e6T^{2} \)
47 \( 1 + (930. - 537. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (330. - 571. i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-2.52e3 - 1.45e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-1.75e3 + 1.01e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (435. - 753. i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 3.71e3T + 2.54e7T^{2} \)
73 \( 1 + (3.21e3 + 1.85e3i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (1.20e3 + 2.08e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 2.66e3iT - 4.74e7T^{2} \)
89 \( 1 + (1.11e4 - 6.43e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 9.16e3iT - 8.85e7T^{2} \)
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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

−14.83649820804773787233033079012, −14.06474428365126523331574631164, −13.07144254077776816623478711643, −11.37950680443580004252976749270, −9.803865065742891607950472021098, −9.087814595392427859756527160214, −7.21942930369303447086615328041, −5.87418290369072958241920172275, −4.48833151356170947645268183568, −2.00347642907202051593560492374, 2.17078508988253219268372227169, 3.27177247839075060633493930092, 5.61996735590469686049577601533, 7.41026877035862646337411509531, 8.504322166752610891949209111171, 10.34151985765139092776708509802, 11.14743413181598073256302086276, 12.80871792497960488471288084018, 13.42843944616152080952009203015, 14.35884521288608657130146618634

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