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Properties

Label 2-50-5.2-c2-0-1
Degree $2$
Conductor $50$
Sign $0.973 - 0.229i$
Analytic cond. $1.36240$
Root an. cond. $1.16721$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s + (2 − 2i)3-s + 2i·4-s + 4·6-s + (−2 − 2i)7-s + (−2 + 2i)8-s + i·9-s − 8·11-s + (4 + 4i)12-s + (−3 + 3i)13-s − 4i·14-s − 4·16-s + (−7 − 7i)17-s + (−1 + i)18-s − 20i·19-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.666 − 0.666i)3-s + 0.5i·4-s + 0.666·6-s + (−0.285 − 0.285i)7-s + (−0.250 + 0.250i)8-s + 0.111i·9-s − 0.727·11-s + (0.333 + 0.333i)12-s + (−0.230 + 0.230i)13-s − 0.285i·14-s − 0.250·16-s + (−0.411 − 0.411i)17-s + (−0.0555 + 0.0555i)18-s − 1.05i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(1.36240\)
Root analytic conductor: \(1.16721\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :1),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.57708 + 0.183625i\)
\(L(\frac12)\) \(\approx\) \(1.57708 + 0.183625i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
5 \( 1 \)
good3 \( 1 + (-2 + 2i)T - 9iT^{2} \)
7 \( 1 + (2 + 2i)T + 49iT^{2} \)
11 \( 1 + 8T + 121T^{2} \)
13 \( 1 + (3 - 3i)T - 169iT^{2} \)
17 \( 1 + (7 + 7i)T + 289iT^{2} \)
19 \( 1 + 20iT - 361T^{2} \)
23 \( 1 + (-2 + 2i)T - 529iT^{2} \)
29 \( 1 - 40iT - 841T^{2} \)
31 \( 1 - 52T + 961T^{2} \)
37 \( 1 + (-3 - 3i)T + 1.36e3iT^{2} \)
41 \( 1 + 8T + 1.68e3T^{2} \)
43 \( 1 + (-42 + 42i)T - 1.84e3iT^{2} \)
47 \( 1 + (-18 - 18i)T + 2.20e3iT^{2} \)
53 \( 1 + (53 - 53i)T - 2.80e3iT^{2} \)
59 \( 1 + 20iT - 3.48e3T^{2} \)
61 \( 1 + 48T + 3.72e3T^{2} \)
67 \( 1 + (62 + 62i)T + 4.48e3iT^{2} \)
71 \( 1 + 28T + 5.04e3T^{2} \)
73 \( 1 + (-47 + 47i)T - 5.32e3iT^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + (18 - 18i)T - 6.88e3iT^{2} \)
89 \( 1 - 80iT - 7.92e3T^{2} \)
97 \( 1 + (-63 - 63i)T + 9.40e3iT^{2} \)
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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

−15.28251355163527999351529236127, −13.96559784724240370666428747266, −13.39879035640600768202794593998, −12.34589087151033955625952130667, −10.74851921991389014260018770078, −8.993375708837880510963674875133, −7.73418527168254582498586886194, −6.74497640504222998871576364165, −4.87141704188521884423827638192, −2.76410783190052260108059071542, 2.82808612201739024620436862174, 4.32055384926547141523222318228, 6.05578039413026976077565028111, 8.151566743344155438981684917321, 9.552185531707097370396717968568, 10.41594705560454574578346185499, 11.92153701163236305548920019596, 13.02918218158221418071367170676, 14.19363308808109727198566100549, 15.21317250705393067168465808194

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