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Properties

Label 2-7e2-7.4-c5-0-6
Degree $2$
Conductor $49$
Sign $0.386 - 0.922i$
Analytic cond. $7.85880$
Root an. cond. $2.80335$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.54 + 4.40i)2-s + (1.04 − 1.80i)3-s + (3.08 − 5.33i)4-s + (20.9 + 36.2i)5-s + 10.5·6-s + 193.·8-s + (119. + 206. i)9-s + (−106. + 184. i)10-s + (−36.0 + 62.4i)11-s + (−6.42 − 11.1i)12-s + 632.·13-s + 87.1·15-s + (394. + 683. i)16-s + (−987. + 1.71e3i)17-s + (−606. + 1.05e3i)18-s + (−932. − 1.61e3i)19-s + ⋯
L(s)  = 1  + (0.449 + 0.778i)2-s + (0.0668 − 0.115i)3-s + (0.0963 − 0.166i)4-s + (0.374 + 0.647i)5-s + 0.120·6-s + 1.07·8-s + (0.491 + 0.850i)9-s + (−0.336 + 0.582i)10-s + (−0.0898 + 0.155i)11-s + (−0.0128 − 0.0222i)12-s + 1.03·13-s + 0.0999·15-s + (0.385 + 0.667i)16-s + (−0.829 + 1.43i)17-s + (−0.441 + 0.764i)18-s + (−0.592 − 1.02i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.10075 + 1.39738i\)
\(L(\frac12)\) \(\approx\) \(2.10075 + 1.39738i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-2.54 - 4.40i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (-1.04 + 1.80i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-20.9 - 36.2i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (36.0 - 62.4i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 632.T + 3.71e5T^{2} \)
17 \( 1 + (987. - 1.71e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (932. + 1.61e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (206. + 358. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 731.T + 2.05e7T^{2} \)
31 \( 1 + (-3.06e3 + 5.30e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (5.17e3 + 8.96e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 3.52e3T + 1.15e8T^{2} \)
43 \( 1 + 1.45e4T + 1.47e8T^{2} \)
47 \( 1 + (1.07e4 + 1.85e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (6.28e3 - 1.08e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.80e4 - 3.12e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-2.01e3 - 3.48e3i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (7.78e3 - 1.34e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 1.21e4T + 1.80e9T^{2} \)
73 \( 1 + (-9.79e3 + 1.69e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (1.80e4 + 3.12e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 - 2.45e4T + 3.93e9T^{2} \)
89 \( 1 + (3.51e4 + 6.08e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 1.05e5T + 8.58e9T^{2} \)
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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

−14.93365228816076403214317533576, −13.71868224907229460274551273155, −13.07809086528291928573776272818, −10.97899464633359070178226596264, −10.34603709605286060085667572856, −8.385647486806063797195804112182, −6.95407357816179491720774032584, −6.04137353694648318862739066194, −4.42340518587421473559970038019, −1.98644084621116462479050701623, 1.40278866562064436716573662027, 3.33795486831397664598392759623, 4.75151172207197844095623963964, 6.65091234187826200771144484833, 8.413298656081512932898341684510, 9.723001854745340391712813007435, 11.06417650414665068550057499994, 12.16743925462537127358952332114, 13.07245726664317252606937943883, 13.96930667248140472370507235915

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