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Properties

Label 2-50-5.3-c10-0-4
Degree $2$
Conductor $50$
Sign $-0.973 - 0.229i$
Analytic cond. $31.7678$
Root an. cond. $5.63629$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−16 + 16i)2-s + (183 + 183i)3-s − 512i·4-s − 5.85e3·6-s + (8.40e3 − 8.40e3i)7-s + (8.19e3 + 8.19e3i)8-s + 7.92e3i·9-s − 1.73e5·11-s + (9.36e4 − 9.36e4i)12-s + (2.32e5 + 2.32e5i)13-s + 2.69e5i·14-s − 2.62e5·16-s + (−1.88e6 + 1.88e6i)17-s + (−1.26e5 − 1.26e5i)18-s + 1.10e6i·19-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (0.753 + 0.753i)3-s − 0.5i·4-s − 0.753·6-s + (0.500 − 0.500i)7-s + (0.250 + 0.250i)8-s + 0.134i·9-s − 1.07·11-s + (0.376 − 0.376i)12-s + (0.626 + 0.626i)13-s + 0.500i·14-s − 0.250·16-s + (−1.32 + 1.32i)17-s + (−0.0671 − 0.0671i)18-s + 0.444i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(31.7678\)
Root analytic conductor: \(5.63629\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :5),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.143602 + 1.23334i\)
\(L(\frac12)\) \(\approx\) \(0.143602 + 1.23334i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (16 - 16i)T \)
5 \( 1 \)
good3 \( 1 + (-183 - 183i)T + 5.90e4iT^{2} \)
7 \( 1 + (-8.40e3 + 8.40e3i)T - 2.82e8iT^{2} \)
11 \( 1 + 1.73e5T + 2.59e10T^{2} \)
13 \( 1 + (-2.32e5 - 2.32e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (1.88e6 - 1.88e6i)T - 2.01e12iT^{2} \)
19 \( 1 - 1.10e6iT - 6.13e12T^{2} \)
23 \( 1 + (-5.22e6 - 5.22e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 2.47e7iT - 4.20e14T^{2} \)
31 \( 1 + 1.00e7T + 8.19e14T^{2} \)
37 \( 1 + (5.63e7 - 5.63e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 1.53e8T + 1.34e16T^{2} \)
43 \( 1 + (5.93e7 + 5.93e7i)T + 2.16e16iT^{2} \)
47 \( 1 + (1.72e8 - 1.72e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (1.96e8 + 1.96e8i)T + 1.74e17iT^{2} \)
59 \( 1 + 6.94e8iT - 5.11e17T^{2} \)
61 \( 1 - 9.06e8T + 7.13e17T^{2} \)
67 \( 1 + (-9.62e8 + 9.62e8i)T - 1.82e18iT^{2} \)
71 \( 1 + 3.12e9T + 3.25e18T^{2} \)
73 \( 1 + (-6.36e8 - 6.36e8i)T + 4.29e18iT^{2} \)
79 \( 1 - 1.96e9iT - 9.46e18T^{2} \)
83 \( 1 + (-5.18e9 - 5.18e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 7.77e9iT - 3.11e19T^{2} \)
97 \( 1 + (-6.40e8 + 6.40e8i)T - 7.37e19iT^{2} \)
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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

−14.20202441323477869611842811378, −13.09654871060494590678292258017, −11.10348688389912512689822227690, −10.23037168459794632868891795677, −8.933975261680247088951979971520, −8.155330902262100489768835735176, −6.66526684595077504458530549256, −4.91591143443582241471298408654, −3.55553304912261514737172320092, −1.65330491118359086232578687574, 0.39064361537725747153155014773, 2.04213215205862413606045956512, 2.87391672545095872862968306295, 5.00725808903218995723362522734, 7.03630951262378005652797742800, 8.177780861911328173698421895512, 8.924261681454050549564019218840, 10.54777454240976708799712707789, 11.60482062712891968878456768252, 13.05455922947514320394314936140

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