Properties

Label 2-50-25.22-c2-0-0
Degree $2$
Conductor $50$
Sign $0.814 - 0.580i$
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  + (−1.26 + 0.642i)2-s + (0.185 − 0.0294i)3-s + (1.17 − 1.61i)4-s + (4.66 + 1.78i)5-s + (−0.215 + 0.156i)6-s + (7.34 + 7.34i)7-s + (−0.442 + 2.79i)8-s + (−8.52 + 2.77i)9-s + (−7.03 + 0.745i)10-s + (5.47 − 16.8i)11-s + (0.170 − 0.335i)12-s + (7.48 + 3.81i)13-s + (−13.9 − 4.53i)14-s + (0.919 + 0.194i)15-s + (−1.23 − 3.80i)16-s + (−29.4 − 4.66i)17-s + ⋯
L(s)  = 1  + (−0.630 + 0.321i)2-s + (0.0619 − 0.00980i)3-s + (0.293 − 0.404i)4-s + (0.933 + 0.357i)5-s + (−0.0358 + 0.0260i)6-s + (1.04 + 1.04i)7-s + (−0.0553 + 0.349i)8-s + (−0.947 + 0.307i)9-s + (−0.703 + 0.0745i)10-s + (0.497 − 1.53i)11-s + (0.0142 − 0.0279i)12-s + (0.575 + 0.293i)13-s + (−0.997 − 0.324i)14-s + (0.0613 + 0.0129i)15-s + (−0.0772 − 0.237i)16-s + (−1.73 − 0.274i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.814 - 0.580i$
Analytic conductor: \(1.36240\)
Root analytic conductor: \(1.16721\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 50,\ (\ :1),\ 0.814 - 0.580i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.962746 + 0.308204i\)
\(L(\frac12)\) \(\approx\) \(0.962746 + 0.308204i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 - 0.642i)T \)
5 \( 1 + (-4.66 - 1.78i)T \)
good3 \( 1 + (-0.185 + 0.0294i)T + (8.55 - 2.78i)T^{2} \)
7 \( 1 + (-7.34 - 7.34i)T + 49iT^{2} \)
11 \( 1 + (-5.47 + 16.8i)T + (-97.8 - 71.1i)T^{2} \)
13 \( 1 + (-7.48 - 3.81i)T + (99.3 + 136. i)T^{2} \)
17 \( 1 + (29.4 + 4.66i)T + (274. + 89.3i)T^{2} \)
19 \( 1 + (9.36 + 12.8i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + (3.14 + 6.17i)T + (-310. + 427. i)T^{2} \)
29 \( 1 + (13.3 - 18.3i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-13.0 + 9.49i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-0.504 + 0.991i)T + (-804. - 1.10e3i)T^{2} \)
41 \( 1 + (14.4 + 44.5i)T + (-1.35e3 + 988. i)T^{2} \)
43 \( 1 + (-8.20 + 8.20i)T - 1.84e3iT^{2} \)
47 \( 1 + (5.42 + 34.2i)T + (-2.10e3 + 682. i)T^{2} \)
53 \( 1 + (31.2 - 4.94i)T + (2.67e3 - 868. i)T^{2} \)
59 \( 1 + (-58.9 + 19.1i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (19.9 - 61.4i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + (59.8 + 9.47i)T + (4.26e3 + 1.38e3i)T^{2} \)
71 \( 1 + (-66.2 - 48.1i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-23.7 - 46.6i)T + (-3.13e3 + 4.31e3i)T^{2} \)
79 \( 1 + (-1.37 + 1.88i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (15.3 - 96.8i)T + (-6.55e3 - 2.12e3i)T^{2} \)
89 \( 1 + (6.69 + 2.17i)T + (6.40e3 + 4.65e3i)T^{2} \)
97 \( 1 + (1.05 + 6.63i)T + (-8.94e3 + 2.90e3i)T^{2} \)
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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

−15.42671578244935805892946223228, −14.33621927591682998720178613530, −13.55480698153396241757134031367, −11.40938180009772184374562985924, −10.95299490712534070447769827200, −8.855915693437316199853336095561, −8.659579301926652454019246430118, −6.47483268513067390497204147896, −5.43676744394116600913056904679, −2.32062181821523018607636021205, 1.79509629060796734913336638619, 4.45353424039347814005113981638, 6.47080268465897249385728005333, 8.069122637956690779811632193474, 9.241150225978815578101761171396, 10.43596438465923174567038574834, 11.47741725940908896917024223843, 12.91193526499473770288148138730, 14.05065332965705803275868514206, 15.11316731600636825336805838872

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