Properties

Label 2-5e2-25.11-c1-0-0
Degree $2$
Conductor $25$
Sign $0.992 - 0.125i$
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  + (−0.5 + 0.363i)2-s + (0.309 − 0.951i)3-s + (−0.5 + 1.53i)4-s + (−1.80 − 1.31i)5-s + (0.190 + 0.587i)6-s − 1.61·7-s + (−0.690 − 2.12i)8-s + (1.61 + 1.17i)9-s + 1.38·10-s + (0.618 − 0.449i)11-s + (1.30 + 0.951i)12-s + (3.92 + 2.85i)13-s + (0.809 − 0.587i)14-s + (−1.80 + 1.31i)15-s + (−1.49 − 1.08i)16-s + (−0.236 − 0.726i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.256i)2-s + (0.178 − 0.549i)3-s + (−0.250 + 0.769i)4-s + (−0.809 − 0.587i)5-s + (0.0779 + 0.239i)6-s − 0.611·7-s + (−0.244 − 0.751i)8-s + (0.539 + 0.391i)9-s + 0.437·10-s + (0.186 − 0.135i)11-s + (0.377 + 0.274i)12-s + (1.08 + 0.791i)13-s + (0.216 − 0.157i)14-s + (−0.467 + 0.339i)15-s + (−0.374 − 0.272i)16-s + (−0.0572 − 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.992 - 0.125i$
Analytic conductor: \(0.199626\)
Root analytic conductor: \(0.446795\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :1/2),\ 0.992 - 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.529534 + 0.0333154i\)
\(L(\frac12)\) \(\approx\) \(0.529534 + 0.0333154i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.80 + 1.31i)T \)
good2 \( 1 + (0.5 - 0.363i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.309 + 0.951i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + 1.61T + 7T^{2} \)
11 \( 1 + (-0.618 + 0.449i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.92 - 2.85i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.236 + 0.726i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.80 + 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (6.66 - 4.84i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.427 - 1.31i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.927 + 2.85i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.42 + 2.48i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.23 - 3.07i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 1.85T + 43T^{2} \)
47 \( 1 + (0.5 - 1.53i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.69 + 5.20i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-3.35 - 2.43i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.80 + 2.76i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.85 - 8.78i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (1.35 - 4.16i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.28 + 5.29i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.954 + 2.93i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.545 + 1.67i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (7.23 - 5.25i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-0.881 + 2.71i)T + (-78.4 - 57.0i)T^{2} \)
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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

−17.73694771594256913984512259039, −16.28658582747629304639915398031, −15.83299224852732678802708324411, −13.57080781772427432924421874687, −12.78745959025090485470809083746, −11.51714826958993390227905514135, −9.275463700894892957584831283175, −8.117834134407304095731906220012, −6.88842797735002891028014004884, −4.01305883699761872997695967334, 3.87632635392220914344699637256, 6.27578904102493561363949704096, 8.400323665832239832022596326221, 9.970660453849164957529609063316, 10.73364131395871901679656565556, 12.41568399753153466766064232053, 14.23329304054160060706842680390, 15.27296137164702920299758227524, 16.11870739004231735436644807772, 18.09369935029835365181006296283

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