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Properties

Label 2-5e2-5.2-c34-0-49
Degree $2$
Conductor $25$
Sign $-0.850 - 0.525i$
Analytic cond. $183.064$
Root an. cond. $13.5301$
Motivic weight $34$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.52e4 + 8.52e4i)2-s + (1.68e8 − 1.68e8i)3-s − 2.62e9i·4-s + 2.86e13·6-s + (−2.04e14 − 2.04e14i)7-s + (1.68e15 − 1.68e15i)8-s − 3.98e16i·9-s − 4.22e17·11-s + (−4.42e17 − 4.42e17i)12-s + (−8.97e18 + 8.97e18i)13-s − 3.49e19i·14-s + 2.43e20·16-s + (2.34e20 + 2.34e20i)17-s + (3.39e21 − 3.39e21i)18-s − 3.96e21i·19-s + ⋯
L(s)  = 1  + (0.650 + 0.650i)2-s + (1.30 − 1.30i)3-s − 0.153i·4-s + 1.69·6-s + (−0.881 − 0.881i)7-s + (0.750 − 0.750i)8-s − 2.38i·9-s − 0.836·11-s + (−0.199 − 0.199i)12-s + (−1.03 + 1.03i)13-s − 1.14i·14-s + 0.823·16-s + (0.283 + 0.283i)17-s + (1.55 − 1.55i)18-s − 0.724i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(183.064\)
Root analytic conductor: \(13.5301\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :17),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(2.041665857\)
\(L(\frac12)\) \(\approx\) \(2.041665857\)
\(L(18)\) 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 \( 1 + (-8.52e4 - 8.52e4i)T + 1.71e10iT^{2} \)
3 \( 1 + (-1.68e8 + 1.68e8i)T - 1.66e16iT^{2} \)
7 \( 1 + (2.04e14 + 2.04e14i)T + 5.41e28iT^{2} \)
11 \( 1 + 4.22e17T + 2.55e35T^{2} \)
13 \( 1 + (8.97e18 - 8.97e18i)T - 7.48e37iT^{2} \)
17 \( 1 + (-2.34e20 - 2.34e20i)T + 6.84e41iT^{2} \)
19 \( 1 + 3.96e21iT - 3.00e43T^{2} \)
23 \( 1 + (-1.87e22 + 1.87e22i)T - 1.98e46iT^{2} \)
29 \( 1 - 3.22e24iT - 5.26e49T^{2} \)
31 \( 1 - 3.15e25T + 5.08e50T^{2} \)
37 \( 1 + (3.27e26 + 3.27e26i)T + 2.08e53iT^{2} \)
41 \( 1 + 9.81e26T + 6.83e54T^{2} \)
43 \( 1 + (2.79e27 - 2.79e27i)T - 3.45e55iT^{2} \)
47 \( 1 + (-9.12e27 - 9.12e27i)T + 7.10e56iT^{2} \)
53 \( 1 + (2.57e28 - 2.57e28i)T - 4.22e58iT^{2} \)
59 \( 1 - 7.57e29iT - 1.61e60T^{2} \)
61 \( 1 + 3.27e30T + 5.02e60T^{2} \)
67 \( 1 + (3.29e30 + 3.29e30i)T + 1.22e62iT^{2} \)
71 \( 1 + 1.48e31T + 8.76e62T^{2} \)
73 \( 1 + (-4.68e31 + 4.68e31i)T - 2.25e63iT^{2} \)
79 \( 1 - 5.17e31iT - 3.30e64T^{2} \)
83 \( 1 + (1.93e32 - 1.93e32i)T - 1.77e65iT^{2} \)
89 \( 1 + 4.72e32iT - 1.90e66T^{2} \)
97 \( 1 + (-5.55e33 - 5.55e33i)T + 3.55e67iT^{2} \)
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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

−10.24566953300208590426184403899, −9.181254439484766304162578641542, −7.70791486306705822107368601488, −7.03397672735180686593657177926, −6.34856008765533121063688349237, −4.66909284403417466001614155034, −3.43185469128584534938278102018, −2.39117066108986294860022659250, −1.24457716201255203319554841104, −0.21880372934803094723111252694, 2.17548634144931358054381722325, 2.92624475283529055914211357577, 3.31825321532471770655728071069, 4.61858538883387481141795717202, 5.43187738454559281939751521623, 7.72236487098478777140055459895, 8.527190027190570785589700084891, 9.828040308108819612260833301874, 10.39487249414172372247536388340, 12.04040185279048343160172741520

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