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Properties

Label 2-5e2-5.4-c9-0-9
Degree $2$
Conductor $25$
Sign $-0.894 + 0.447i$
Analytic cond. $12.8758$
Root an. cond. $3.58829$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 31.7i·2-s − 268. i·3-s − 498.·4-s + 8.53e3·6-s − 637. i·7-s + 440. i·8-s − 5.24e4·9-s − 4.90e4·11-s + 1.33e5i·12-s + 7.27e4i·13-s + 2.02e4·14-s − 2.69e5·16-s − 6.73e4i·17-s − 1.66e6i·18-s − 3.41e5·19-s + ⋯
L(s)  = 1  + 1.40i·2-s − 1.91i·3-s − 0.972·4-s + 2.68·6-s − 0.100i·7-s + 0.0380i·8-s − 2.66·9-s − 1.00·11-s + 1.86i·12-s + 0.706i·13-s + 0.140·14-s − 1.02·16-s − 0.195i·17-s − 3.74i·18-s − 0.600·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(12.8758\)
Root analytic conductor: \(3.58829\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :9/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.0346821 - 0.146915i\)
\(L(\frac12)\) \(\approx\) \(0.0346821 - 0.146915i\)
\(L(\frac{11}{2})\) 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 \( 1 - 31.7iT - 512T^{2} \)
3 \( 1 + 268. iT - 1.96e4T^{2} \)
7 \( 1 + 637. iT - 4.03e7T^{2} \)
11 \( 1 + 4.90e4T + 2.35e9T^{2} \)
13 \( 1 - 7.27e4iT - 1.06e10T^{2} \)
17 \( 1 + 6.73e4iT - 1.18e11T^{2} \)
19 \( 1 + 3.41e5T + 3.22e11T^{2} \)
23 \( 1 + 1.34e5iT - 1.80e12T^{2} \)
29 \( 1 + 4.45e6T + 1.45e13T^{2} \)
31 \( 1 - 4.56e5T + 2.64e13T^{2} \)
37 \( 1 + 1.30e7iT - 1.29e14T^{2} \)
41 \( 1 + 2.56e7T + 3.27e14T^{2} \)
43 \( 1 - 3.42e6iT - 5.02e14T^{2} \)
47 \( 1 - 3.39e7iT - 1.11e15T^{2} \)
53 \( 1 + 8.42e7iT - 3.29e15T^{2} \)
59 \( 1 - 7.46e7T + 8.66e15T^{2} \)
61 \( 1 - 1.78e8T + 1.16e16T^{2} \)
67 \( 1 + 6.94e7iT - 2.72e16T^{2} \)
71 \( 1 + 2.07e8T + 4.58e16T^{2} \)
73 \( 1 + 3.02e8iT - 5.88e16T^{2} \)
79 \( 1 + 3.72e8T + 1.19e17T^{2} \)
83 \( 1 - 4.50e8iT - 1.86e17T^{2} \)
89 \( 1 + 5.82e7T + 3.50e17T^{2} \)
97 \( 1 + 7.85e8iT - 7.60e17T^{2} \)
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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

−14.79847090827018661687956385394, −13.76063378276915221487759568926, −12.84917786249603060583835253227, −11.37898192043253522665937850547, −8.640830357817648931087717845766, −7.60005597542704149604360686815, −6.73398100316321392349052534716, −5.52797801513244824277603784294, −2.14874760932246224980062611409, −0.05926775966929461258012590304, 2.70919388528199052686863164702, 3.90691337546193740276913612751, 5.30567983365970725671355808252, 8.652466870589776206896540441459, 10.01379446238856697084403833050, 10.56445655419732009325689612672, 11.65949065605490719835746315481, 13.26063417008591885474441325273, 14.99217891762836059937887354354, 15.82860274610069648886754000270

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