Properties

Label 2-7e2-7.2-c3-0-6
Degree $2$
Conductor $49$
Sign $-0.605 + 0.795i$
Analytic cond. $2.89109$
Root an. cond. $1.70032$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.5 − 4.33i)2-s + (−8.50 − 14.7i)4-s − 45.0·8-s + (13.5 − 23.3i)9-s + (34 + 58.8i)11-s + (−44.5 + 77.0i)16-s + (−67.5 − 116. i)18-s + 340·22-s + (20 − 34.6i)23-s + (62.5 + 108. i)25-s − 166·29-s + (42.5 + 73.6i)32-s − 459.·36-s + (−225 + 389. i)37-s − 180·43-s + (578. − 1.00e3i)44-s + ⋯
L(s)  = 1  + (0.883 − 1.53i)2-s + (−1.06 − 1.84i)4-s − 1.98·8-s + (0.5 − 0.866i)9-s + (0.931 + 1.61i)11-s + (−0.695 + 1.20i)16-s + (−0.883 − 1.53i)18-s + 3.29·22-s + (0.181 − 0.314i)23-s + (0.5 + 0.866i)25-s − 1.06·29-s + (0.234 + 0.406i)32-s − 2.12·36-s + (−0.999 + 1.73i)37-s − 0.638·43-s + (1.98 − 3.43i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.884737 - 1.78472i\)
\(L(\frac12)\) \(\approx\) \(0.884737 - 1.78472i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.5 + 4.33i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-34 - 58.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-20 + 34.6i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 166T + 2.43e4T^{2} \)
31 \( 1 + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (225 - 389. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 + 180T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (295 + 510. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-370 - 640. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 688T + 3.57e5T^{2} \)
73 \( 1 + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-692 + 1.19e3i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 9.12e5T^{2} \)
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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

−14.56790425228937279716035732933, −13.14577009457773151276223702368, −12.35158545687795217665964664544, −11.52360495984211484578701326346, −10.09827857541056548882226540943, −9.306031528502591944840417068787, −6.80492720372833074655131100255, −4.84578728785744800676254557824, −3.61191549279543295011614079246, −1.61408681898428496179850998824, 3.79768215730231864999750578192, 5.32295841663858738679933007277, 6.51814749140058946600446650259, 7.79471782090051441495854058209, 8.935841680855088529441486642920, 10.99618450099550377412016358504, 12.57030356794099798553145416739, 13.71392684831262013852724372012, 14.26829920098738522370876216374, 15.58150961034859080593446467880

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