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Properties

Label 2-50-5.4-c3-0-0
Degree $2$
Conductor $50$
Sign $-0.894 - 0.447i$
Analytic cond. $2.95009$
Root an. cond. $1.71758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 8i·3-s − 4·4-s − 16·6-s − 4i·7-s − 8i·8-s − 37·9-s + 12·11-s − 32i·12-s + 58i·13-s + 8·14-s + 16·16-s + 66i·17-s − 74i·18-s + 100·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.53i·3-s − 0.5·4-s − 1.08·6-s − 0.215i·7-s − 0.353i·8-s − 1.37·9-s + 0.328·11-s − 0.769i·12-s + 1.23i·13-s + 0.152·14-s + 0.250·16-s + 0.941i·17-s − 0.968i·18-s + 1.20·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(2.95009\)
Root analytic conductor: \(1.71758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.282444 + 1.19645i\)
\(L(\frac12)\) \(\approx\) \(0.282444 + 1.19645i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 \)
good3 \( 1 - 8iT - 27T^{2} \)
7 \( 1 + 4iT - 343T^{2} \)
11 \( 1 - 12T + 1.33e3T^{2} \)
13 \( 1 - 58iT - 2.19e3T^{2} \)
17 \( 1 - 66iT - 4.91e3T^{2} \)
19 \( 1 - 100T + 6.85e3T^{2} \)
23 \( 1 + 132iT - 1.21e4T^{2} \)
29 \( 1 - 90T + 2.43e4T^{2} \)
31 \( 1 - 152T + 2.97e4T^{2} \)
37 \( 1 + 34iT - 5.06e4T^{2} \)
41 \( 1 + 438T + 6.89e4T^{2} \)
43 \( 1 + 32iT - 7.95e4T^{2} \)
47 \( 1 + 204iT - 1.03e5T^{2} \)
53 \( 1 + 222iT - 1.48e5T^{2} \)
59 \( 1 + 420T + 2.05e5T^{2} \)
61 \( 1 - 902T + 2.26e5T^{2} \)
67 \( 1 + 1.02e3iT - 3.00e5T^{2} \)
71 \( 1 - 432T + 3.57e5T^{2} \)
73 \( 1 + 362iT - 3.89e5T^{2} \)
79 \( 1 - 160T + 4.93e5T^{2} \)
83 \( 1 + 72iT - 5.71e5T^{2} \)
89 \( 1 + 810T + 7.04e5T^{2} \)
97 \( 1 - 1.10e3iT - 9.12e5T^{2} \)
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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

−15.62001423636497566063866823441, −14.66146576740618367689496819192, −13.78049441861810918113040198402, −11.92747077283982877643766860601, −10.52235850261201176709979784220, −9.539024558288012235209398976354, −8.471655985474746939248355619300, −6.60120719230790750030694868445, −4.96052845392361014241777925406, −3.82831637849933163936845337038, 1.05760840811300091664909075802, 2.90361989337505064412224784537, 5.50750450250675981569748446203, 7.15420894931494698984089348467, 8.288851680221354422659785850911, 9.832569946823623660128707259755, 11.51417460820323294099251633669, 12.21457394514825895843208794355, 13.31271593880255605271715297694, 14.00096104293718566486892224623

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