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Properties

Label 2-50-5.3-c12-0-13
Degree $2$
Conductor $50$
Sign $0.973 + 0.229i$
Analytic cond. $45.6996$
Root an. cond. $6.76015$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−32 + 32i)2-s + (748. + 748. i)3-s − 2.04e3i·4-s − 4.79e4·6-s + (1.21e5 − 1.21e5i)7-s + (6.55e4 + 6.55e4i)8-s + 5.89e5i·9-s − 8.55e5·11-s + (1.53e6 − 1.53e6i)12-s + (−5.63e5 − 5.63e5i)13-s + 7.74e6i·14-s − 4.19e6·16-s + (1.33e7 − 1.33e7i)17-s + (−1.88e7 − 1.88e7i)18-s − 9.07e7i·19-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (1.02 + 1.02i)3-s − 0.5i·4-s − 1.02·6-s + (1.02 − 1.02i)7-s + (0.250 + 0.250i)8-s + 1.10i·9-s − 0.482·11-s + (0.513 − 0.513i)12-s + (−0.116 − 0.116i)13-s + 1.02i·14-s − 0.250·16-s + (0.554 − 0.554i)17-s + (−0.554 − 0.554i)18-s − 1.92i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(45.6996\)
Root analytic conductor: \(6.76015\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :6),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.178168958\)
\(L(\frac12)\) \(\approx\) \(2.178168958\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (32 - 32i)T \)
5 \( 1 \)
good3 \( 1 + (-748. - 748. i)T + 5.31e5iT^{2} \)
7 \( 1 + (-1.21e5 + 1.21e5i)T - 1.38e10iT^{2} \)
11 \( 1 + 8.55e5T + 3.13e12T^{2} \)
13 \( 1 + (5.63e5 + 5.63e5i)T + 2.32e13iT^{2} \)
17 \( 1 + (-1.33e7 + 1.33e7i)T - 5.82e14iT^{2} \)
19 \( 1 + 9.07e7iT - 2.21e15T^{2} \)
23 \( 1 + (6.22e7 + 6.22e7i)T + 2.19e16iT^{2} \)
29 \( 1 + 9.61e8iT - 3.53e17T^{2} \)
31 \( 1 + 1.55e9T + 7.87e17T^{2} \)
37 \( 1 + (7.41e8 - 7.41e8i)T - 6.58e18iT^{2} \)
41 \( 1 - 9.28e8T + 2.25e19T^{2} \)
43 \( 1 + (-2.23e9 - 2.23e9i)T + 3.99e19iT^{2} \)
47 \( 1 + (-7.22e9 + 7.22e9i)T - 1.16e20iT^{2} \)
53 \( 1 + (-9.05e8 - 9.05e8i)T + 4.91e20iT^{2} \)
59 \( 1 + 1.35e10iT - 1.77e21T^{2} \)
61 \( 1 - 6.94e10T + 2.65e21T^{2} \)
67 \( 1 + (1.13e11 - 1.13e11i)T - 8.18e21iT^{2} \)
71 \( 1 - 1.09e11T + 1.64e22T^{2} \)
73 \( 1 + (-8.58e9 - 8.58e9i)T + 2.29e22iT^{2} \)
79 \( 1 - 1.74e11iT - 5.90e22T^{2} \)
83 \( 1 + (-2.87e11 - 2.87e11i)T + 1.06e23iT^{2} \)
89 \( 1 + 2.51e11iT - 2.46e23T^{2} \)
97 \( 1 + (2.97e11 - 2.97e11i)T - 6.93e23iT^{2} \)
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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

−13.46662400594209811033884839541, −11.25197061116816448580576163400, −10.28553978757166963442562503004, −9.274772527696183791956683801634, −8.171388218007912678188698065969, −7.24398742836646459552438068066, −5.10228617993318397162240225926, −4.07477294724975816556806974815, −2.44205169695443463157956886411, −0.60923654570945186390706323494, 1.50865961299634322263185184718, 2.05853611807583513964056498902, 3.45534061250252217518276201118, 5.57567763366499733375388040887, 7.49998638951817758642643491568, 8.198147011965248916043734422842, 9.096671497513203932125217266732, 10.66567793021022017095217870401, 12.12825664619157294149496972073, 12.70550418866276725504916807905

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