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Properties

Label 2-5e2-5.3-c32-0-5
Degree $2$
Conductor $25$
Sign $0.850 - 0.525i$
Analytic cond. $162.166$
Root an. cond. $12.7344$
Motivic weight $32$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.35e4 − 7.35e4i)2-s + (4.29e6 + 4.29e6i)3-s − 6.51e9i·4-s + 6.31e11·6-s + (−1.36e13 + 1.36e13i)7-s + (−1.63e14 − 1.63e14i)8-s − 1.81e15i·9-s − 1.07e16·11-s + (2.79e16 − 2.79e16i)12-s + (−5.33e17 − 5.33e17i)13-s + 2.00e18i·14-s + 3.95e18·16-s + (−1.80e19 + 1.80e19i)17-s + (−1.33e20 − 1.33e20i)18-s + 4.66e20i·19-s + ⋯
L(s)  = 1  + (1.12 − 1.12i)2-s + (0.0997 + 0.0997i)3-s − 1.51i·4-s + 0.223·6-s + (−0.410 + 0.410i)7-s + (−0.580 − 0.580i)8-s − 0.980i·9-s − 0.234·11-s + (0.151 − 0.151i)12-s + (−0.801 − 0.801i)13-s + 0.921i·14-s + 0.214·16-s + (−0.370 + 0.370i)17-s + (−1.09 − 1.09i)18-s + 1.61i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.850 - 0.525i$
Analytic conductor: \(162.166\)
Root analytic conductor: \(12.7344\)
Motivic weight: \(32\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :16),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.741115075\)
\(L(\frac12)\) \(\approx\) \(1.741115075\)
\(L(17)\) 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 \( 1 + (-7.35e4 + 7.35e4i)T - 4.29e9iT^{2} \)
3 \( 1 + (-4.29e6 - 4.29e6i)T + 1.85e15iT^{2} \)
7 \( 1 + (1.36e13 - 1.36e13i)T - 1.10e27iT^{2} \)
11 \( 1 + 1.07e16T + 2.11e33T^{2} \)
13 \( 1 + (5.33e17 + 5.33e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (1.80e19 - 1.80e19i)T - 2.36e39iT^{2} \)
19 \( 1 - 4.66e20iT - 8.31e40T^{2} \)
23 \( 1 + (7.85e20 + 7.85e20i)T + 3.76e43iT^{2} \)
29 \( 1 + 2.31e23iT - 6.26e46T^{2} \)
31 \( 1 - 3.24e23T + 5.29e47T^{2} \)
37 \( 1 + (4.94e24 - 4.94e24i)T - 1.52e50iT^{2} \)
41 \( 1 + 1.24e26T + 4.06e51T^{2} \)
43 \( 1 + (-1.83e25 - 1.83e25i)T + 1.86e52iT^{2} \)
47 \( 1 + (3.45e26 - 3.45e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (-3.32e27 - 3.32e27i)T + 1.50e55iT^{2} \)
59 \( 1 - 3.29e28iT - 4.64e56T^{2} \)
61 \( 1 - 1.89e28T + 1.35e57T^{2} \)
67 \( 1 + (-1.76e29 + 1.76e29i)T - 2.71e58iT^{2} \)
71 \( 1 + 1.94e29T + 1.73e59T^{2} \)
73 \( 1 + (3.08e29 + 3.08e29i)T + 4.22e59iT^{2} \)
79 \( 1 - 3.31e30iT - 5.29e60T^{2} \)
83 \( 1 + (-6.01e30 - 6.01e30i)T + 2.57e61iT^{2} \)
89 \( 1 - 2.42e31iT - 2.40e62T^{2} \)
97 \( 1 + (2.36e31 - 2.36e31i)T - 3.77e63iT^{2} \)
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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

−11.97869453761319983702077541121, −10.46724003174287074950461233446, −9.697750816464547229180171160518, −8.107188030113421355329719058432, −6.31442814075746121790106874409, −5.35645224256963169012714534118, −4.09739966366346001045153342156, −3.22923495243249509605630364214, −2.33897853840315875306179473959, −1.11422261654518462222127088485, 0.21156736250235107528229648648, 2.07340075701690028126992018026, 3.31662730286911785503189001228, 4.64929982555652020264842608164, 5.19039942482876336790499693159, 6.79730550984160956138321331238, 7.19372708551469297553351130419, 8.576049701706468234248902839522, 10.13517125494270496097244055023, 11.60861726622645321081694835623

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