↑

Properties

Label 2-5e2-5.4-c37-0-24
Degree $2$
Conductor $25$
Sign $0.447 + 0.894i$
Analytic cond. $216.785$
Root an. cond. $14.7236$
Motivic weight $37$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.86e5i·2-s − 2.05e7i·3-s + 5.56e10·4-s − 5.89e12·6-s − 1.97e15i·7-s − 5.52e16i·8-s + 4.49e17·9-s − 2.57e19·11-s − 1.14e18i·12-s + 5.42e20i·13-s − 5.64e20·14-s − 8.14e21·16-s + 3.52e22i·17-s − 1.28e23i·18-s − 6.82e23·19-s + ⋯
L(s)  = 1  − 0.771i·2-s − 0.0306i·3-s + 0.404·4-s − 0.0236·6-s − 0.458i·7-s − 1.08i·8-s + 0.999·9-s − 1.39·11-s − 0.0124i·12-s + 1.33i·13-s − 0.353·14-s − 0.431·16-s + 0.608i·17-s − 0.770i·18-s − 1.50·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(216.785\)
Root analytic conductor: \(14.7236\)
Motivic weight: \(37\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :37/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(19)\) \(\approx\) \(2.587616929\)
\(L(\frac12)\) \(\approx\) \(2.587616929\)
\(L(\frac{39}{2})\) 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 \( 1 + 2.86e5iT - 1.37e11T^{2} \)
3 \( 1 + 2.05e7iT - 4.50e17T^{2} \)
7 \( 1 + 1.97e15iT - 1.85e31T^{2} \)
11 \( 1 + 2.57e19T + 3.40e38T^{2} \)
13 \( 1 - 5.42e20iT - 1.64e41T^{2} \)
17 \( 1 - 3.52e22iT - 3.36e45T^{2} \)
19 \( 1 + 6.82e23T + 2.06e47T^{2} \)
23 \( 1 + 8.19e22iT - 2.42e50T^{2} \)
29 \( 1 - 1.51e27T + 1.28e54T^{2} \)
31 \( 1 - 2.60e27T + 1.51e55T^{2} \)
37 \( 1 - 1.30e29iT - 1.05e58T^{2} \)
41 \( 1 + 4.07e29T + 4.70e59T^{2} \)
43 \( 1 + 2.92e30iT - 2.74e60T^{2} \)
47 \( 1 + 3.58e30iT - 7.37e61T^{2} \)
53 \( 1 - 3.56e31iT - 6.28e63T^{2} \)
59 \( 1 - 3.03e32T + 3.32e65T^{2} \)
61 \( 1 - 1.16e33T + 1.14e66T^{2} \)
67 \( 1 - 2.44e33iT - 3.67e67T^{2} \)
71 \( 1 - 6.30e33T + 3.13e68T^{2} \)
73 \( 1 - 1.02e34iT - 8.76e68T^{2} \)
79 \( 1 + 1.20e35T + 1.63e70T^{2} \)
83 \( 1 + 3.26e35iT - 1.01e71T^{2} \)
89 \( 1 - 1.56e36T + 1.34e72T^{2} \)
97 \( 1 - 1.07e36iT - 3.24e73T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.44792068257538995765520063614, −10.18667744503749556051297117913, −8.502603576357806633103941424335, −7.16963935013912798873156370159, −6.41847882068679130062088845174, −4.65586780171470725972535602826, −3.82011241451543731876518533666, −2.46072723556122825635893930327, −1.75245722670016953137950360283, −0.65371497003403686505607208548, 0.64145346317658265134020937135, 2.15497441414689432035231039873, 2.90568439056820371127755326393, 4.65368637811279987078930293727, 5.55419398780132498861449836878, 6.60893528521541672257309134465, 7.69986559906273678470105243679, 8.407627401336255002951176270681, 10.10210048092420164387390040526, 10.88110899299806706398653234803

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