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Properties

Label 2-7e2-7.4-c5-0-3
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  + (−5.5 − 9.52i)2-s + (−44.5 + 77.0i)4-s + 626.·8-s + (121.5 + 210. i)9-s + (38 − 65.8i)11-s + (−2.02e3 − 3.50e3i)16-s + (1.33e3 − 2.31e3i)18-s − 836·22-s + (2.47e3 + 4.28e3i)23-s + (1.56e3 − 2.70e3i)25-s + 7.28e3·29-s + (−1.22e4 + 2.11e4i)32-s − 2.16e4·36-s + (4.44e3 + 7.69e3i)37-s + 1.17e4·43-s + (3.38e3 + 5.85e3i)44-s + ⋯
L(s)  = 1  + (−0.972 − 1.68i)2-s + (−1.39 + 2.40i)4-s + 3.46·8-s + (0.5 + 0.866i)9-s + (0.0946 − 0.164i)11-s + (−1.97 − 3.42i)16-s + (0.972 − 1.68i)18-s − 0.368·22-s + (0.975 + 1.69i)23-s + (0.5 − 0.866i)25-s + 1.60·29-s + (−2.11 + 3.65i)32-s − 2.78·36-s + (0.533 + 0.924i)37-s + 0.968·43-s + (0.263 + 0.456i)44-s + ⋯

Functional equation

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

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: $\chi_{49} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 49,\ (\ :5/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.810875 - 0.401972i\)
\(L(\frac12)\) \(\approx\) \(0.810875 - 0.401972i\)
\(L(\frac{7}{2})\) 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 \( 1 + (5.5 + 9.52i)T + (-16 + 27.7i)T^{2} \)
3 \( 1 + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-38 + 65.8i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 3.71e5T^{2} \)
17 \( 1 + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.47e3 - 4.28e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 - 7.28e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.44e3 - 7.69e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 1.15e8T^{2} \)
43 \( 1 - 1.17e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (1.22e4 - 2.12e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (3.46e4 - 6.00e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 2.22e3T + 1.80e9T^{2} \)
73 \( 1 + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (4.00e4 + 6.94e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 3.93e9T^{2} \)
89 \( 1 + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 8.58e9T^{2} \)
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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

−13.83402098173615237309563946479, −12.94075601727643739658634060184, −11.80890450617666156216612863444, −10.79188271225877721964154660048, −9.889417980539586984598260024936, −8.676409011842994005936267099199, −7.50592385933956759888849705418, −4.54420280589619847263599005074, −2.85586637449605103085036914909, −1.23887419323159435713920577928, 0.817019250532297601689552575162, 4.68982984388865013993011455444, 6.28676626806682348573626347119, 7.17909803497092025660594220749, 8.571915459378809512632589738251, 9.499418099516848113232982554769, 10.66600418153605608773764310597, 12.76032305916006270488256223925, 14.25487887236242164915534054361, 15.02803088170051434083531658119

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