Properties

Label 2-5e3-125.109-c1-0-6
Degree $2$
Conductor $125$
Sign $0.584 + 0.811i$
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  + (0.161 − 1.28i)2-s + (−1.71 + 1.41i)3-s + (0.321 + 0.0826i)4-s + (0.720 − 2.11i)5-s + (1.54 + 2.42i)6-s + (2.27 − 0.740i)7-s + (1.10 − 2.80i)8-s + (0.365 − 1.91i)9-s + (−2.59 − 1.26i)10-s + (4.57 + 0.577i)11-s + (−0.668 + 0.314i)12-s + (−5.66 − 1.08i)13-s + (−0.579 − 3.04i)14-s + (1.76 + 4.65i)15-s + (−2.82 − 1.55i)16-s + (1.07 + 4.20i)17-s + ⋯
L(s)  = 1  + (0.114 − 0.905i)2-s + (−0.990 + 0.819i)3-s + (0.160 + 0.0413i)4-s + (0.322 − 0.946i)5-s + (0.628 + 0.990i)6-s + (0.861 − 0.279i)7-s + (0.391 − 0.990i)8-s + (0.121 − 0.639i)9-s + (−0.820 − 0.400i)10-s + (1.37 + 0.174i)11-s + (−0.193 + 0.0908i)12-s + (−1.57 − 0.299i)13-s + (−0.155 − 0.812i)14-s + (0.456 + 1.20i)15-s + (−0.706 − 0.388i)16-s + (0.261 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(0.998130\)
Root analytic conductor: \(0.999064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{125} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 125,\ (\ :1/2),\ 0.584 + 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.942389 - 0.482437i\)
\(L(\frac12)\) \(\approx\) \(0.942389 - 0.482437i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.720 + 2.11i)T \)
good2 \( 1 + (-0.161 + 1.28i)T + (-1.93 - 0.497i)T^{2} \)
3 \( 1 + (1.71 - 1.41i)T + (0.562 - 2.94i)T^{2} \)
7 \( 1 + (-2.27 + 0.740i)T + (5.66 - 4.11i)T^{2} \)
11 \( 1 + (-4.57 - 0.577i)T + (10.6 + 2.73i)T^{2} \)
13 \( 1 + (5.66 + 1.08i)T + (12.0 + 4.78i)T^{2} \)
17 \( 1 + (-1.07 - 4.20i)T + (-14.8 + 8.18i)T^{2} \)
19 \( 1 + (3.02 - 3.65i)T + (-3.56 - 18.6i)T^{2} \)
23 \( 1 + (0.551 - 0.587i)T + (-1.44 - 22.9i)T^{2} \)
29 \( 1 + (-0.0122 - 0.193i)T + (-28.7 + 3.63i)T^{2} \)
31 \( 1 + (-0.935 + 0.240i)T + (27.1 - 14.9i)T^{2} \)
37 \( 1 + (3.98 - 7.25i)T + (-19.8 - 31.2i)T^{2} \)
41 \( 1 + (-0.442 + 0.415i)T + (2.57 - 40.9i)T^{2} \)
43 \( 1 + (-6.60 - 9.09i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (1.47 + 3.71i)T + (-34.2 + 32.1i)T^{2} \)
53 \( 1 + (6.95 + 4.41i)T + (22.5 + 47.9i)T^{2} \)
59 \( 1 + (-2.33 - 4.97i)T + (-37.6 + 45.4i)T^{2} \)
61 \( 1 + (8.65 + 8.12i)T + (3.83 + 60.8i)T^{2} \)
67 \( 1 + (1.08 + 0.0683i)T + (66.4 + 8.39i)T^{2} \)
71 \( 1 + (-0.401 + 0.158i)T + (51.7 - 48.6i)T^{2} \)
73 \( 1 + (-13.2 - 6.24i)T + (46.5 + 56.2i)T^{2} \)
79 \( 1 + (-0.594 - 0.719i)T + (-14.8 + 77.6i)T^{2} \)
83 \( 1 + (1.41 + 1.16i)T + (15.5 + 81.5i)T^{2} \)
89 \( 1 + (1.00 - 2.14i)T + (-56.7 - 68.5i)T^{2} \)
97 \( 1 + (-7.38 + 0.464i)T + (96.2 - 12.1i)T^{2} \)
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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

−12.64418703586149893730781672012, −12.09407025004541848572275462354, −11.27162547353864256733997704083, −10.25832437716582669569148528709, −9.602740016565053076677434955970, −7.979846420112522931950843668446, −6.29456592126853041151922418259, −4.88797171240222863843897217179, −4.08602891010138108790204847377, −1.63735507293336118339154437175, 2.15079917470006800446872239184, 4.94752054847000029153263369753, 6.06333413544489132673517375990, 6.93257146661650073870580881260, 7.47046710068769731439178665052, 9.209969063253051866305073802028, 10.84416285042962272219274222604, 11.57895673384517032972717007900, 12.24223430747878409066426172153, 14.01385359179837907587226364493

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