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Properties

Label 2-5e3-25.16-c1-0-0
Degree $2$
Conductor $125$
Sign $0.525 + 0.850i$
Analytic cond. $0.998130$
Root an. cond. $0.999064$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.86 − 1.35i)2-s + (0.146 + 0.451i)3-s + (1.02 + 3.16i)4-s + (0.338 − 1.04i)6-s + 3.03·7-s + (0.951 − 2.92i)8-s + (2.24 − 1.63i)9-s + (−1.61 − 1.17i)11-s + (−1.28 + 0.930i)12-s + (1.15 − 0.838i)13-s + (−5.67 − 4.12i)14-s + (−0.357 + 0.259i)16-s + (0.574 − 1.76i)17-s − 6.40·18-s + (0.279 − 0.859i)19-s + ⋯
L(s)  = 1  + (−1.32 − 0.959i)2-s + (0.0847 + 0.260i)3-s + (0.514 + 1.58i)4-s + (0.138 − 0.425i)6-s + 1.14·7-s + (0.336 − 1.03i)8-s + (0.748 − 0.543i)9-s + (−0.487 − 0.354i)11-s + (−0.369 + 0.268i)12-s + (0.320 − 0.232i)13-s + (−1.51 − 1.10i)14-s + (−0.0893 + 0.0649i)16-s + (0.139 − 0.429i)17-s − 1.51·18-s + (0.0640 − 0.197i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.583822 - 0.325694i\)
\(L(\frac12)\) \(\approx\) \(0.583822 - 0.325694i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
good2 \( 1 + (1.86 + 1.35i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.146 - 0.451i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 3.03T + 7T^{2} \)
11 \( 1 + (1.61 + 1.17i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.15 + 0.838i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.574 + 1.76i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.279 + 0.859i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.69 - 1.95i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.22 - 3.76i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.99 - 6.12i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.09 - 2.24i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.48 + 1.07i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.59T + 43T^{2} \)
47 \( 1 + (1.48 + 4.56i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (2.93 + 9.03i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (8.61 - 6.25i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (11.5 + 8.39i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (3.30 - 10.1i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-3.85 - 11.8i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.216 + 0.157i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.64 + 8.15i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.89 - 12.0i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-3.85 - 2.80i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.07 + 9.47i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.85467782743974378219305474701, −11.80035583501735246073891826949, −10.94724972211039140703683537774, −10.19210826744538142001256223924, −9.088808118168410683212108552838, −8.268317078993769631595166447932, −7.16898766113012006580071429755, −5.04948242638157859878706389168, −3.25290137108682544449570886973, −1.42292266719503744380162412498, 1.67656832144337952264212292492, 4.68821393585128151383170852692, 6.17387252566215980315408848753, 7.53493615223447595513538734296, 7.944549432773240180170847538703, 9.091918703268995093787829543976, 10.26022195822747889584054398529, 11.06408727336207604529402844091, 12.56680073512995316293414607761, 13.80554596448669206869967703938

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