Properties

Label 2-5e2-25.4-c5-0-3
Degree $2$
Conductor $25$
Sign $-0.168 - 0.985i$
Analytic cond. $4.00959$
Root an. cond. $2.00239$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.71 + 1.85i)2-s + (−10.7 + 14.8i)3-s + (3.32 + 2.41i)4-s + (37.8 + 41.1i)5-s + (−89.2 + 64.8i)6-s + 177. i·7-s + (−98.5 − 135. i)8-s + (−28.9 − 89.2i)9-s + (140. + 305. i)10-s + (174. − 536. i)11-s + (−71.8 + 23.3i)12-s + (578. − 188. i)13-s + (−328. + 1.01e3i)14-s + (−1.01e3 + 118. i)15-s + (−351. − 1.08e3i)16-s + (950. + 1.30e3i)17-s + ⋯
L(s)  = 1  + (1.01 + 0.328i)2-s + (−0.692 + 0.952i)3-s + (0.104 + 0.0755i)4-s + (0.677 + 0.735i)5-s + (−1.01 + 0.735i)6-s + 1.36i·7-s + (−0.544 − 0.748i)8-s + (−0.119 − 0.367i)9-s + (0.443 + 0.965i)10-s + (0.434 − 1.33i)11-s + (−0.143 + 0.0467i)12-s + (0.949 − 0.308i)13-s + (−0.448 + 1.38i)14-s + (−1.16 + 0.136i)15-s + (−0.343 − 1.05i)16-s + (0.797 + 1.09i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.168 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.168 - 0.985i$
Analytic conductor: \(4.00959\)
Root analytic conductor: \(2.00239\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :5/2),\ -0.168 - 0.985i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.26364 + 1.49792i\)
\(L(\frac12)\) \(\approx\) \(1.26364 + 1.49792i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-37.8 - 41.1i)T \)
good2 \( 1 + (-5.71 - 1.85i)T + (25.8 + 18.8i)T^{2} \)
3 \( 1 + (10.7 - 14.8i)T + (-75.0 - 231. i)T^{2} \)
7 \( 1 - 177. iT - 1.68e4T^{2} \)
11 \( 1 + (-174. + 536. i)T + (-1.30e5 - 9.46e4i)T^{2} \)
13 \( 1 + (-578. + 188. i)T + (3.00e5 - 2.18e5i)T^{2} \)
17 \( 1 + (-950. - 1.30e3i)T + (-4.38e5 + 1.35e6i)T^{2} \)
19 \( 1 + (-1.26e3 + 921. i)T + (7.65e5 - 2.35e6i)T^{2} \)
23 \( 1 + (1.77e3 + 576. i)T + (5.20e6 + 3.78e6i)T^{2} \)
29 \( 1 + (450. + 326. i)T + (6.33e6 + 1.95e7i)T^{2} \)
31 \( 1 + (4.54e3 - 3.30e3i)T + (8.84e6 - 2.72e7i)T^{2} \)
37 \( 1 + (-7.49e3 + 2.43e3i)T + (5.61e7 - 4.07e7i)T^{2} \)
41 \( 1 + (144. + 444. i)T + (-9.37e7 + 6.80e7i)T^{2} \)
43 \( 1 + 5.24e3iT - 1.47e8T^{2} \)
47 \( 1 + (-1.28e4 + 1.77e4i)T + (-7.08e7 - 2.18e8i)T^{2} \)
53 \( 1 + (-6.14e3 + 8.46e3i)T + (-1.29e8 - 3.97e8i)T^{2} \)
59 \( 1 + (-3.19e3 - 9.82e3i)T + (-5.78e8 + 4.20e8i)T^{2} \)
61 \( 1 + (-1.09e3 + 3.35e3i)T + (-6.83e8 - 4.96e8i)T^{2} \)
67 \( 1 + (1.01e4 + 1.39e4i)T + (-4.17e8 + 1.28e9i)T^{2} \)
71 \( 1 + (3.18e4 + 2.31e4i)T + (5.57e8 + 1.71e9i)T^{2} \)
73 \( 1 + (6.93e4 + 2.25e4i)T + (1.67e9 + 1.21e9i)T^{2} \)
79 \( 1 + (2.98e4 + 2.16e4i)T + (9.50e8 + 2.92e9i)T^{2} \)
83 \( 1 + (-1.05e4 - 1.45e4i)T + (-1.21e9 + 3.74e9i)T^{2} \)
89 \( 1 + (-2.68e4 + 8.25e4i)T + (-4.51e9 - 3.28e9i)T^{2} \)
97 \( 1 + (6.88e4 - 9.47e4i)T + (-2.65e9 - 8.16e9i)T^{2} \)
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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

−16.42874076298370639488526195123, −15.49747589201732971959968470180, −14.52013879825626476315036333063, −13.34453033317282617982868177348, −11.73394625313864265463445115929, −10.45789039698263352957878131224, −9.030104257862499302842838033848, −5.96813438960283356421980462357, −5.60463006301962853000511171157, −3.49301965078223867741540585249, 1.29639921694876882136815720655, 4.22981008743864673048740789497, 5.82429626805141716760159840347, 7.42050637809059335108815010369, 9.651496996617440347060530185451, 11.58555235179376159567089578584, 12.54599975838956440030948379636, 13.44629298544413607329842110604, 14.25651214768108861373071940186, 16.49139680175616534549024951533

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