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Properties

Label 2-5e2-5.4-c35-0-3
Degree $2$
Conductor $25$
Sign $-0.447 - 0.894i$
Analytic cond. $193.987$
Root an. cond. $13.9279$
Motivic weight $35$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.65e5i·2-s − 3.45e8i·3-s + 7.09e9·4-s + 5.71e13·6-s + 1.25e14i·7-s + 6.84e15i·8-s − 6.96e16·9-s − 1.70e18·11-s − 2.45e18i·12-s − 4.94e19i·13-s − 2.06e19·14-s − 8.86e20·16-s − 1.32e21i·17-s − 1.14e22i·18-s − 3.94e21·19-s + ⋯
L(s)  = 1  + 0.890i·2-s − 1.54i·3-s + 0.206·4-s + 1.37·6-s + 0.203i·7-s + 1.07i·8-s − 1.39·9-s − 1.01·11-s − 0.319i·12-s − 1.58i·13-s − 0.181·14-s − 0.750·16-s − 0.387i·17-s − 1.23i·18-s − 0.165·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(193.987\)
Root analytic conductor: \(13.9279\)
Motivic weight: \(35\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :35/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(18)\) \(\approx\) \(0.5529248503\)
\(L(\frac12)\) \(\approx\) \(0.5529248503\)
\(L(\frac{37}{2})\) 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 \( 1 - 1.65e5iT - 3.43e10T^{2} \)
3 \( 1 + 3.45e8iT - 5.00e16T^{2} \)
7 \( 1 - 1.25e14iT - 3.78e29T^{2} \)
11 \( 1 + 1.70e18T + 2.81e36T^{2} \)
13 \( 1 + 4.94e19iT - 9.72e38T^{2} \)
17 \( 1 + 1.32e21iT - 1.16e43T^{2} \)
19 \( 1 + 3.94e21T + 5.70e44T^{2} \)
23 \( 1 + 3.48e23iT - 4.57e47T^{2} \)
29 \( 1 + 3.21e25T + 1.52e51T^{2} \)
31 \( 1 - 3.41e25T + 1.57e52T^{2} \)
37 \( 1 - 4.03e27iT - 7.71e54T^{2} \)
41 \( 1 - 8.65e27T + 2.80e56T^{2} \)
43 \( 1 - 9.89e26iT - 1.48e57T^{2} \)
47 \( 1 + 1.95e29iT - 3.33e58T^{2} \)
53 \( 1 + 9.96e29iT - 2.23e60T^{2} \)
59 \( 1 + 3.91e30T + 9.54e61T^{2} \)
61 \( 1 - 7.64e29T + 3.06e62T^{2} \)
67 \( 1 - 1.64e32iT - 8.17e63T^{2} \)
71 \( 1 - 7.65e31T + 6.22e64T^{2} \)
73 \( 1 + 7.08e32iT - 1.64e65T^{2} \)
79 \( 1 + 2.34e33T + 2.61e66T^{2} \)
83 \( 1 - 5.18e33iT - 1.47e67T^{2} \)
89 \( 1 + 1.44e34T + 1.69e68T^{2} \)
97 \( 1 - 2.87e34iT - 3.44e69T^{2} \)
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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

−11.69975328143724385985308445300, −10.44059042746309076302849535650, −8.418859581883484977166711592739, −7.80436939588658847394906633484, −7.00099937497097080898121769407, −5.96227514690010511325612756000, −5.20195962628852298957484091267, −2.89709274652995206591141693777, −2.20814603213805941606285415939, −0.921911153825830818527602097055, 0.10333780655279679341571362778, 1.67440156398831818804705498507, 2.72601412830201847859763483370, 3.81872192799924787795160296745, 4.46948676542373090284987413026, 5.83122117297054241882544134864, 7.33001080827591021177124613070, 9.031161655450162730082585522002, 9.834044233814358165476788920743, 10.76957775765709858486827075511

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