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Properties

Label 2-50-5.4-c25-0-35
Degree $2$
Conductor $50$
Sign $-0.894 - 0.447i$
Analytic cond. $197.998$
Root an. cond. $14.0711$
Motivic weight $25$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.09e3i·2-s − 9.79e4i·3-s − 1.67e7·4-s − 4.01e8·6-s − 4.08e10i·7-s + 6.87e10i·8-s + 8.37e11·9-s − 1.45e13·11-s + 1.64e12i·12-s − 8.78e13i·13-s − 1.67e14·14-s + 2.81e14·16-s − 2.65e15i·17-s − 3.43e15i·18-s + 1.39e16·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.106i·3-s − 0.5·4-s − 0.0752·6-s − 1.11i·7-s + 0.353i·8-s + 0.988·9-s − 1.39·11-s + 0.0532i·12-s − 1.04i·13-s − 0.789·14-s + 0.250·16-s − 1.10i·17-s − 0.699i·18-s + 1.45·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(197.998\)
Root analytic conductor: \(14.0711\)
Motivic weight: \(25\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :25/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(13)\) \(\approx\) \(1.894744156\)
\(L(\frac12)\) \(\approx\) \(1.894744156\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4.09e3iT \)
5 \( 1 \)
good3 \( 1 + 9.79e4iT - 8.47e11T^{2} \)
7 \( 1 + 4.08e10iT - 1.34e21T^{2} \)
11 \( 1 + 1.45e13T + 1.08e26T^{2} \)
13 \( 1 + 8.78e13iT - 7.05e27T^{2} \)
17 \( 1 + 2.65e15iT - 5.77e30T^{2} \)
19 \( 1 - 1.39e16T + 9.30e31T^{2} \)
23 \( 1 + 8.58e16iT - 1.10e34T^{2} \)
29 \( 1 + 2.08e18T + 3.63e36T^{2} \)
31 \( 1 - 2.66e18T + 1.92e37T^{2} \)
37 \( 1 + 5.13e19iT - 1.60e39T^{2} \)
41 \( 1 - 2.33e20T + 2.08e40T^{2} \)
43 \( 1 - 4.01e19iT - 6.86e40T^{2} \)
47 \( 1 - 2.79e20iT - 6.34e41T^{2} \)
53 \( 1 + 4.25e20iT - 1.27e43T^{2} \)
59 \( 1 - 8.33e21T + 1.86e44T^{2} \)
61 \( 1 - 2.42e22T + 4.29e44T^{2} \)
67 \( 1 + 1.24e23iT - 4.48e45T^{2} \)
71 \( 1 + 9.30e22T + 1.91e46T^{2} \)
73 \( 1 + 4.04e22iT - 3.82e46T^{2} \)
79 \( 1 - 8.05e23T + 2.75e47T^{2} \)
83 \( 1 + 8.98e21iT - 9.48e47T^{2} \)
89 \( 1 + 3.55e24T + 5.42e48T^{2} \)
97 \( 1 + 8.66e24iT - 4.66e49T^{2} \)
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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

−10.32247826475391347518613669076, −9.539517750479977680472315560784, −7.80631201524660919398038165262, −7.29687387727108583570474859522, −5.47219486260854562763706443049, −4.50878727406498061928764319524, −3.34887640491460949794846702058, −2.34856068818202039546226208422, −0.900776489494010525656916243899, −0.43595442275936098591851901586, 1.25973068833346783059988508179, 2.43454575719537163880212140940, 3.83540640228395727319647498258, 5.05635649389781934395349442503, 5.84700779839146239820219516712, 7.14274003614780968625282130884, 8.074120073125140872709118209134, 9.241095860485609669822128988065, 10.10886845509996706379191077376, 11.56910619768844341456790390458

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