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Properties

Label 2-5e2-5.4-c7-0-6
Degree $2$
Conductor $25$
Sign $0.894 + 0.447i$
Analytic cond. $7.80962$
Root an. cond. $2.79457$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 20.2i·2-s − 45.4i·3-s − 281.·4-s + 920.·6-s − 1.36e3i·7-s − 3.10e3i·8-s + 118.·9-s − 1.01e3·11-s + 1.28e4i·12-s − 3.64e3i·13-s + 2.77e4·14-s + 2.68e4·16-s − 5.53e3i·17-s + 2.40e3i·18-s − 2.32e4·19-s + ⋯
L(s)  = 1  + 1.78i·2-s − 0.972i·3-s − 2.19·4-s + 1.73·6-s − 1.50i·7-s − 2.14i·8-s + 0.0544·9-s − 0.229·11-s + 2.13i·12-s − 0.459i·13-s + 2.70·14-s + 1.63·16-s − 0.273i·17-s + 0.0973i·18-s − 0.777·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(7.80962\)
Root analytic conductor: \(2.79457\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :7/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.05694 - 0.249511i\)
\(L(\frac12)\) \(\approx\) \(1.05694 - 0.249511i\)
\(L(\frac{9}{2})\) 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 \( 1 - 20.2iT - 128T^{2} \)
3 \( 1 + 45.4iT - 2.18e3T^{2} \)
7 \( 1 + 1.36e3iT - 8.23e5T^{2} \)
11 \( 1 + 1.01e3T + 1.94e7T^{2} \)
13 \( 1 + 3.64e3iT - 6.27e7T^{2} \)
17 \( 1 + 5.53e3iT - 4.10e8T^{2} \)
19 \( 1 + 2.32e4T + 8.93e8T^{2} \)
23 \( 1 + 9.63e4iT - 3.40e9T^{2} \)
29 \( 1 - 1.39e5T + 1.72e10T^{2} \)
31 \( 1 + 2.08e5T + 2.75e10T^{2} \)
37 \( 1 - 4.48e5iT - 9.49e10T^{2} \)
41 \( 1 + 7.97e4T + 1.94e11T^{2} \)
43 \( 1 + 2.27e5iT - 2.71e11T^{2} \)
47 \( 1 - 7.19e5iT - 5.06e11T^{2} \)
53 \( 1 - 4.95e5iT - 1.17e12T^{2} \)
59 \( 1 - 8.10e4T + 2.48e12T^{2} \)
61 \( 1 - 3.15e6T + 3.14e12T^{2} \)
67 \( 1 - 1.06e6iT - 6.06e12T^{2} \)
71 \( 1 + 3.00e6T + 9.09e12T^{2} \)
73 \( 1 + 4.05e5iT - 1.10e13T^{2} \)
79 \( 1 - 4.71e6T + 1.92e13T^{2} \)
83 \( 1 + 5.14e6iT - 2.71e13T^{2} \)
89 \( 1 - 8.62e6T + 4.42e13T^{2} \)
97 \( 1 + 6.99e6iT - 8.07e13T^{2} \)
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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

−16.07739114946974975484605654799, −14.63641497529048986197299525962, −13.63580503053231979968966077727, −12.79593095884880378651022397154, −10.31061411355093051988697602026, −8.325690002440526509303951178074, −7.29449016687830167159137973475, −6.45665563683092410430485017931, −4.50130115838290016174306038995, −0.58187380381659014864298974099, 2.07044693710879193344715879886, 3.70408314139548339755187416311, 5.22423735668896154356071698587, 8.822083004067304349406172288272, 9.667447775981312450124025627429, 10.88810469940143606798078943168, 11.98026224603421751026197280408, 13.05763522996617342990543085082, 14.75117640559909360652544604101, 15.95058298940236550879742194400

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