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Properties

Label 2-50-5.2-c6-0-5
Degree $2$
Conductor $50$
Sign $0.229 + 0.973i$
Analytic cond. $11.5027$
Root an. cond. $3.39156$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4 − 4i)2-s + (−3 + 3i)3-s + 32i·4-s + 24·6-s + (−117 − 117i)7-s + (128 − 128i)8-s + 711i·9-s + 972·11-s + (−96 − 96i)12-s + (2.41e3 − 2.41e3i)13-s + 936i·14-s − 1.02e3·16-s + (−4.81e3 − 4.81e3i)17-s + (2.84e3 − 2.84e3i)18-s − 5.74e3i·19-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.111 + 0.111i)3-s + 0.5i·4-s + 0.111·6-s + (−0.341 − 0.341i)7-s + (0.250 − 0.250i)8-s + 0.975i·9-s + 0.730·11-s + (−0.0555 − 0.0555i)12-s + (1.09 − 1.09i)13-s + 0.341i·14-s − 0.250·16-s + (−0.979 − 0.979i)17-s + (0.487 − 0.487i)18-s − 0.836i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(11.5027\)
Root analytic conductor: \(3.39156\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :3),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.939652 - 0.743658i\)
\(L(\frac12)\) \(\approx\) \(0.939652 - 0.743658i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4 + 4i)T \)
5 \( 1 \)
good3 \( 1 + (3 - 3i)T - 729iT^{2} \)
7 \( 1 + (117 + 117i)T + 1.17e5iT^{2} \)
11 \( 1 - 972T + 1.77e6T^{2} \)
13 \( 1 + (-2.41e3 + 2.41e3i)T - 4.82e6iT^{2} \)
17 \( 1 + (4.81e3 + 4.81e3i)T + 2.41e7iT^{2} \)
19 \( 1 + 5.74e3iT - 4.70e7T^{2} \)
23 \( 1 + (-1.35e4 + 1.35e4i)T - 1.48e8iT^{2} \)
29 \( 1 + 1.75e4iT - 5.94e8T^{2} \)
31 \( 1 - 2.97e4T + 8.87e8T^{2} \)
37 \( 1 + (-1.54e4 - 1.54e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 9.54e4T + 4.75e9T^{2} \)
43 \( 1 + (9.86e4 - 9.86e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (-6.65e4 - 6.65e4i)T + 1.07e10iT^{2} \)
53 \( 1 + (4.84e4 - 4.84e4i)T - 2.21e10iT^{2} \)
59 \( 1 + 1.43e5iT - 4.21e10T^{2} \)
61 \( 1 + 1.37e5T + 5.15e10T^{2} \)
67 \( 1 + (1.87e5 + 1.87e5i)T + 9.04e10iT^{2} \)
71 \( 1 - 2.55e5T + 1.28e11T^{2} \)
73 \( 1 + (-2.86e5 + 2.86e5i)T - 1.51e11iT^{2} \)
79 \( 1 - 8.34e5iT - 2.43e11T^{2} \)
83 \( 1 + (-4.53e5 + 4.53e5i)T - 3.26e11iT^{2} \)
89 \( 1 + 4.72e5iT - 4.96e11T^{2} \)
97 \( 1 + (4.73e5 + 4.73e5i)T + 8.32e11iT^{2} \)
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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

−13.71270746844075082334705205950, −12.96692722958197900932997544577, −11.35783963609951211133815168085, −10.65126996024579502114791063556, −9.305784059059550310908505515270, −8.086612267571196422519615747277, −6.58352265849248625608203508400, −4.60165453963465215657240520896, −2.79450655974802303024503981702, −0.73627104136058483244756001975, 1.36163771651462571922494868877, 3.87144298440077722852710347875, 6.02306223962358275921165931371, 6.83854252986168900676777869139, 8.679175576450179322988943994485, 9.385937032051428833119117325511, 10.99325996309710594894005271382, 12.13929863571695574507368247838, 13.52655525798405036744170684660, 14.78707594224488561937843289528

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