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Properties

Label 2-5e3-25.11-c1-0-3
Degree $2$
Conductor $125$
Sign $0.526 + 0.850i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.526 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.526 + 0.850i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.526 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58293 - 0.882022i\)
\(L(\frac12)\) \(\approx\) \(1.58293 - 0.882022i\)
\(L(\frac{3}{2})\) 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

−13.11929915097082875510031993134, −12.45050532133126147799590362306, −11.37721254750256987995062106886, −10.24731731000774315635563086033, −9.677314518738723250195404443638, −7.61130565525451402883261522629, −6.11547494431231781371395254198, −4.91303661089280062182410966095, −3.78618392323871594925482783495, −2.40386567053213333965417650351, 3.20362452341818143024499620600, 4.45012197463429773012824138212, 5.91426700662497836953546673554, 6.68085975050181996319782043361, 7.62140112512924906455517114496, 9.240409521309884996960382367225, 10.52590501377522926186587045972, 12.28509488134141291784130608872, 12.66914139146565446379561337632, 13.56249938877005315111782882775

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