Properties

Label 2-7e2-49.11-c1-0-1
Degree $2$
Conductor $49$
Sign $0.436 - 0.899i$
Analytic cond. $0.391266$
Root an. cond. $0.625513$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73 + 1.60i)2-s + (2.78 − 0.419i)3-s + (0.267 − 3.57i)4-s + (−0.567 + 1.44i)5-s + (−4.14 + 5.19i)6-s + (−2.62 − 0.356i)7-s + (2.32 + 2.91i)8-s + (4.68 − 1.44i)9-s + (−1.34 − 3.41i)10-s + (−0.663 − 0.204i)11-s + (−0.752 − 10.0i)12-s + (−0.928 − 4.06i)13-s + (5.11 − 3.59i)14-s + (−0.972 + 4.26i)15-s + (−1.64 − 0.247i)16-s + (−0.659 + 0.449i)17-s + ⋯
L(s)  = 1  + (−1.22 + 1.13i)2-s + (1.60 − 0.241i)3-s + (0.133 − 1.78i)4-s + (−0.253 + 0.647i)5-s + (−1.69 + 2.12i)6-s + (−0.990 − 0.134i)7-s + (0.823 + 1.03i)8-s + (1.56 − 0.482i)9-s + (−0.424 − 1.08i)10-s + (−0.200 − 0.0616i)11-s + (−0.217 − 2.89i)12-s + (−0.257 − 1.12i)13-s + (1.36 − 0.960i)14-s + (−0.251 + 1.10i)15-s + (−0.410 − 0.0618i)16-s + (−0.159 + 0.109i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.436 - 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.436 - 0.899i$
Analytic conductor: \(0.391266\)
Root analytic conductor: \(0.625513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :1/2),\ 0.436 - 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.560802 + 0.351107i\)
\(L(\frac12)\) \(\approx\) \(0.560802 + 0.351107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.62 + 0.356i)T \)
good2 \( 1 + (1.73 - 1.60i)T + (0.149 - 1.99i)T^{2} \)
3 \( 1 + (-2.78 + 0.419i)T + (2.86 - 0.884i)T^{2} \)
5 \( 1 + (0.567 - 1.44i)T + (-3.66 - 3.40i)T^{2} \)
11 \( 1 + (0.663 + 0.204i)T + (9.08 + 6.19i)T^{2} \)
13 \( 1 + (0.928 + 4.06i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + (0.659 - 0.449i)T + (6.21 - 15.8i)T^{2} \)
19 \( 1 + (-1.05 - 1.82i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.52 + 3.08i)T + (8.40 + 21.4i)T^{2} \)
29 \( 1 + (-0.0358 - 0.0172i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (-1.21 + 2.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.720 - 9.60i)T + (-36.5 + 5.51i)T^{2} \)
41 \( 1 + (-2.97 - 3.73i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (4.04 - 5.07i)T + (-9.56 - 41.9i)T^{2} \)
47 \( 1 + (-4.10 + 3.81i)T + (3.51 - 46.8i)T^{2} \)
53 \( 1 + (0.686 - 9.15i)T + (-52.4 - 7.89i)T^{2} \)
59 \( 1 + (3.92 + 10.0i)T + (-43.2 + 40.1i)T^{2} \)
61 \( 1 + (0.356 + 4.75i)T + (-60.3 + 9.09i)T^{2} \)
67 \( 1 + (-3.95 + 6.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.10 - 1.49i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (1.84 + 1.71i)T + (5.45 + 72.7i)T^{2} \)
79 \( 1 + (-5.31 - 9.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.05 + 9.01i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (0.0252 - 0.00779i)T + (73.5 - 50.1i)T^{2} \)
97 \( 1 + 11.7T + 97T^{2} \)
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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

−15.61540502891348357817285480017, −15.08269434613960827461542393619, −14.05638578901500111056730276623, −12.81089068630158204445864256798, −10.33164129482903977107995137610, −9.540850964098852450522454002363, −8.286444571660247181107528293392, −7.56479930757604903470772125851, −6.38363374043148200910775020665, −3.12674182957468222818705875292, 2.37691468357065645303090888648, 3.84102437282300842830156183768, 7.45612688951209375677273072749, 8.777677118844299256822977948482, 9.244883689149721267332635912695, 10.21820045297416062911681638642, 11.90283550754502223596520713579, 12.95104072452204862939015479542, 14.13142175717493926515158384502, 15.70707864299546086693319892371

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