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Properties

Label 2-50-5.4-c11-0-2
Degree $2$
Conductor $50$
Sign $-0.894 + 0.447i$
Analytic cond. $38.4171$
Root an. cond. $6.19815$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 32i·2-s + 745. i·3-s − 1.02e3·4-s + 2.38e4·6-s + 7.14e4i·7-s + 3.27e4i·8-s − 3.78e5·9-s − 3.45e5·11-s − 7.63e5i·12-s + 1.50e6i·13-s + 2.28e6·14-s + 1.04e6·16-s + 5.39e6i·17-s + 1.21e7i·18-s + 1.11e7·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.77i·3-s − 0.5·4-s + 1.25·6-s + 1.60i·7-s + 0.353i·8-s − 2.13·9-s − 0.647·11-s − 0.885i·12-s + 1.12i·13-s + 1.13·14-s + 0.250·16-s + 0.921i·17-s + 1.51i·18-s + 1.03·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(38.4171\)
Root analytic conductor: \(6.19815\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :11/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.240328 - 1.01804i\)
\(L(\frac12)\) \(\approx\) \(0.240328 - 1.01804i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 32iT \)
5 \( 1 \)
good3 \( 1 - 745. iT - 1.77e5T^{2} \)
7 \( 1 - 7.14e4iT - 1.97e9T^{2} \)
11 \( 1 + 3.45e5T + 2.85e11T^{2} \)
13 \( 1 - 1.50e6iT - 1.79e12T^{2} \)
17 \( 1 - 5.39e6iT - 3.42e13T^{2} \)
19 \( 1 - 1.11e7T + 1.16e14T^{2} \)
23 \( 1 + 5.27e6iT - 9.52e14T^{2} \)
29 \( 1 - 1.86e7T + 1.22e16T^{2} \)
31 \( 1 - 7.10e7T + 2.54e16T^{2} \)
37 \( 1 + 3.23e8iT - 1.77e17T^{2} \)
41 \( 1 + 9.11e8T + 5.50e17T^{2} \)
43 \( 1 + 1.16e9iT - 9.29e17T^{2} \)
47 \( 1 + 2.81e8iT - 2.47e18T^{2} \)
53 \( 1 - 4.05e9iT - 9.26e18T^{2} \)
59 \( 1 + 4.89e9T + 3.01e19T^{2} \)
61 \( 1 - 1.07e10T + 4.35e19T^{2} \)
67 \( 1 + 3.70e9iT - 1.22e20T^{2} \)
71 \( 1 - 3.45e9T + 2.31e20T^{2} \)
73 \( 1 + 2.21e10iT - 3.13e20T^{2} \)
79 \( 1 + 7.02e9T + 7.47e20T^{2} \)
83 \( 1 - 5.55e10iT - 1.28e21T^{2} \)
89 \( 1 - 9.29e9T + 2.77e21T^{2} \)
97 \( 1 + 4.71e10iT - 7.15e21T^{2} \)
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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

−13.99830448292698566440189635156, −12.25665570660114406785752868026, −11.37248795467662721994527872286, −10.25529370831909468972350033212, −9.279574880116980799509611916173, −8.543019520393921364632640709745, −5.74915873154231172816765285241, −4.79316006670890400928285490097, −3.47556467703925050798453300776, −2.25881700101800446779129339675, 0.34147618556170258148670321114, 1.12535313185064251576717722925, 3.07731742982426718278088511284, 5.19477420731705977830571333726, 6.65124482350148246988307924686, 7.49434008307918100866338133063, 8.115067441271984359358289477985, 10.08520845354184800385455531280, 11.58059262766115651982256367465, 13.02415444758528316551736186276

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