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Properties

Label 2-5e2-25.9-c1-0-0
Degree $2$
Conductor $25$
Sign $0.188 - 0.982i$
Analytic cond. $0.199626$
Root an. cond. $0.446795$
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  + (−1.35 + 1.86i)2-s + (−0.451 + 0.146i)3-s + (−1.02 − 3.16i)4-s + (2.19 + 0.420i)5-s + (0.338 − 1.04i)6-s − 3.03i·7-s + (2.92 + 0.951i)8-s + (−2.24 + 1.63i)9-s + (−3.76 + 3.53i)10-s + (−1.61 − 1.17i)11-s + (0.930 + 1.28i)12-s + (0.838 + 1.15i)13-s + (5.67 + 4.12i)14-s + (−1.05 + 0.132i)15-s + (−0.357 + 0.259i)16-s + (−1.76 − 0.574i)17-s + ⋯
L(s)  = 1  + (−0.959 + 1.32i)2-s + (−0.260 + 0.0847i)3-s + (−0.514 − 1.58i)4-s + (0.982 + 0.187i)5-s + (0.138 − 0.425i)6-s − 1.14i·7-s + (1.03 + 0.336i)8-s + (−0.748 + 0.543i)9-s + (−1.19 + 1.11i)10-s + (−0.487 − 0.354i)11-s + (0.268 + 0.369i)12-s + (0.232 + 0.320i)13-s + (1.51 + 1.10i)14-s + (−0.272 + 0.0342i)15-s + (−0.0893 + 0.0649i)16-s + (−0.429 − 0.139i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.188 - 0.982i$
Analytic conductor: \(0.199626\)
Root analytic conductor: \(0.446795\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :1/2),\ 0.188 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.338102 + 0.279416i\)
\(L(\frac12)\) \(\approx\) \(0.338102 + 0.279416i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.19 - 0.420i)T \)
good2 \( 1 + (1.35 - 1.86i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.451 - 0.146i)T + (2.42 - 1.76i)T^{2} \)
7 \( 1 + 3.03iT - 7T^{2} \)
11 \( 1 + (1.61 + 1.17i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.838 - 1.15i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.76 + 0.574i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.279 - 0.859i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.95 - 2.69i)T + (-7.10 - 21.8i)T^{2} \)
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 \( 1 + (-2.24 - 3.09i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.48 + 1.07i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 3.59iT - 43T^{2} \)
47 \( 1 + (4.56 - 1.48i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-9.03 + 2.93i)T + (42.8 - 31.1i)T^{2} \)
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 \( 1 + (-10.1 - 3.30i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.85 - 11.8i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.157 + 0.216i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.64 - 8.15i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (12.0 + 3.89i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (3.85 + 2.80i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (9.47 - 3.07i)T + (78.4 - 57.0i)T^{2} \)
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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

−17.52117840616285466591855096200, −16.91057168293796945249074019103, −15.96599924038615834768568340323, −14.34254366238323078089611004746, −13.54259238772985619509116866189, −10.90587930528900373524074703507, −9.810108913165090789065643859250, −8.313327649596639150128601111549, −6.85744330449983430802479407420, −5.52198619498244300122709572032, 2.48425883173590507807753867406, 5.82385696664051564495606249014, 8.587410407659643003356611381504, 9.482819874120175830761398188808, 10.82063917633549326686182765412, 12.06627944208059647097832957114, 13.02837660570818258256266349106, 14.93013691025311420320926652276, 16.79318904475727817106591641484, 18.10654887689206407956242808015

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