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Properties

Label 2-7e2-7.2-c5-0-8
Degree $2$
Conductor $49$
Sign $0.605 - 0.795i$
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  + (1 − 1.73i)2-s + (12.4 + 21.6i)3-s + (14 + 24.2i)4-s + (37.4 − 64.8i)5-s + 49.9·6-s + 120·8-s + (−190. + 329. i)9-s + (−74.9 − 129. i)10-s + (142 + 245. i)11-s + (−349. + 605. i)12-s − 524.·13-s + 1.87e3·15-s + (−328 + 568. i)16-s + (−74.9 − 129. i)17-s + (380. + 659. i)18-s + (1.08e3 − 1.88e3i)19-s + ⋯
L(s)  = 1  + (0.176 − 0.306i)2-s + (0.801 + 1.38i)3-s + (0.437 + 0.757i)4-s + (0.670 − 1.16i)5-s + 0.566·6-s + 0.662·8-s + (−0.783 + 1.35i)9-s + (−0.236 − 0.410i)10-s + (0.353 + 0.612i)11-s + (−0.701 + 1.21i)12-s − 0.860·13-s + 2.14·15-s + (−0.320 + 0.554i)16-s + (−0.0628 − 0.108i)17-s + (0.277 + 0.480i)18-s + (0.690 − 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(7.85880\)
Root analytic conductor: \(2.80335\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{49} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.46425 + 1.22159i\)
\(L(\frac12)\) \(\approx\) \(2.46425 + 1.22159i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-16 - 27.7i)T^{2} \)
3 \( 1 + (-12.4 - 21.6i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-37.4 + 64.8i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-142 - 245. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 524.T + 3.71e5T^{2} \)
17 \( 1 + (74.9 + 129. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-1.08e3 + 1.88e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (748 - 1.29e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 4.36e3T + 2.05e7T^{2} \)
31 \( 1 + (3.22e3 + 5.58e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-6.31e3 + 1.09e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 9.44e3T + 1.15e8T^{2} \)
43 \( 1 + 1.35e3T + 1.47e8T^{2} \)
47 \( 1 + (-5.02e3 + 8.69e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (7.07e3 + 1.22e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.86e4 + 3.23e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.77e4 - 3.08e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.82e3 - 3.15e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 3.56e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.03e4 - 3.53e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-2.73e4 + 4.72e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 524.T + 3.93e9T^{2} \)
89 \( 1 + (1.01e4 - 1.76e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.83e5T + 8.58e9T^{2} \)
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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

−14.86807408809820583461481656668, −13.56469142270406660998701193003, −12.56361942491683022986851865274, −11.19834348342818381303576430817, −9.631971694020178028461560344358, −9.123294581355580247245908769293, −7.59887013950997562054157932418, −5.08946411523801904039923169111, −3.98393248112484017443431254570, −2.32190473032238038213195177085, 1.55636061981345596010102734220, 2.81538228789731591294416239411, 5.92052696889564886452907688310, 6.80744185819402772458189020328, 7.79018681960116744934746544000, 9.609386568902881017510385621093, 10.87671221190772231817002794734, 12.32168739159110605009982245649, 13.80102951264684939445283515678, 14.24728155583565917777151901159

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