Properties

Label 2-5e2-25.16-c5-0-1
Degree $2$
Conductor $25$
Sign $-0.160 - 0.987i$
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  + (2.28 + 1.65i)2-s + (7.91 + 24.3i)3-s + (−7.42 − 22.8i)4-s + (−0.762 + 55.8i)5-s + (−22.3 + 68.7i)6-s − 43.3·7-s + (48.8 − 150. i)8-s + (−334. + 242. i)9-s + (−94.5 + 126. i)10-s + (545. + 396. i)11-s + (497. − 361. i)12-s + (261. − 189. i)13-s + (−99.0 − 71.9i)14-s + (−1.36e3 + 423. i)15-s + (−260. + 189. i)16-s + (648. − 1.99e3i)17-s + ⋯
L(s)  = 1  + (0.403 + 0.293i)2-s + (0.507 + 1.56i)3-s + (−0.232 − 0.714i)4-s + (−0.0136 + 0.999i)5-s + (−0.253 + 0.779i)6-s − 0.334·7-s + (0.270 − 0.831i)8-s + (−1.37 + 0.998i)9-s + (−0.298 + 0.399i)10-s + (1.35 + 0.987i)11-s + (0.997 − 0.725i)12-s + (0.428 − 0.311i)13-s + (−0.135 − 0.0981i)14-s + (−1.56 + 0.486i)15-s + (−0.254 + 0.184i)16-s + (0.544 − 1.67i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $-0.160 - 0.987i$
Analytic conductor: \(4.00959\)
Root analytic conductor: \(2.00239\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :5/2),\ -0.160 - 0.987i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.24929 + 1.46886i\)
\(L(\frac12)\) \(\approx\) \(1.24929 + 1.46886i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.762 - 55.8i)T \)
good2 \( 1 + (-2.28 - 1.65i)T + (9.88 + 30.4i)T^{2} \)
3 \( 1 + (-7.91 - 24.3i)T + (-196. + 142. i)T^{2} \)
7 \( 1 + 43.3T + 1.68e4T^{2} \)
11 \( 1 + (-545. - 396. i)T + (4.97e4 + 1.53e5i)T^{2} \)
13 \( 1 + (-261. + 189. i)T + (1.14e5 - 3.53e5i)T^{2} \)
17 \( 1 + (-648. + 1.99e3i)T + (-1.14e6 - 8.34e5i)T^{2} \)
19 \( 1 + (-317. + 976. i)T + (-2.00e6 - 1.45e6i)T^{2} \)
23 \( 1 + (-1.20e3 - 876. i)T + (1.98e6 + 6.12e6i)T^{2} \)
29 \( 1 + (-654. - 2.01e3i)T + (-1.65e7 + 1.20e7i)T^{2} \)
31 \( 1 + (-319. + 984. i)T + (-2.31e7 - 1.68e7i)T^{2} \)
37 \( 1 + (-4.66e3 + 3.39e3i)T + (2.14e7 - 6.59e7i)T^{2} \)
41 \( 1 + (1.43e4 - 1.04e4i)T + (3.58e7 - 1.10e8i)T^{2} \)
43 \( 1 + 4.14e3T + 1.47e8T^{2} \)
47 \( 1 + (4.08e3 + 1.25e4i)T + (-1.85e8 + 1.34e8i)T^{2} \)
53 \( 1 + (4.88e3 + 1.50e4i)T + (-3.38e8 + 2.45e8i)T^{2} \)
59 \( 1 + (-7.54e3 + 5.48e3i)T + (2.20e8 - 6.79e8i)T^{2} \)
61 \( 1 + (-2.25e3 - 1.64e3i)T + (2.60e8 + 8.03e8i)T^{2} \)
67 \( 1 + (218. - 673. i)T + (-1.09e9 - 7.93e8i)T^{2} \)
71 \( 1 + (2.40e4 + 7.41e4i)T + (-1.45e9 + 1.06e9i)T^{2} \)
73 \( 1 + (-5.59e4 - 4.06e4i)T + (6.40e8 + 1.97e9i)T^{2} \)
79 \( 1 + (-1.17e4 - 3.62e4i)T + (-2.48e9 + 1.80e9i)T^{2} \)
83 \( 1 + (-7.89e3 + 2.43e4i)T + (-3.18e9 - 2.31e9i)T^{2} \)
89 \( 1 + (-6.06e4 - 4.40e4i)T + (1.72e9 + 5.31e9i)T^{2} \)
97 \( 1 + (-4.80e3 - 1.47e4i)T + (-6.94e9 + 5.04e9i)T^{2} \)
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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

−16.35767955212824589127461811428, −15.28423255913559805040963724990, −14.69416932329631154221750255689, −13.76158281396370563704703506781, −11.34373024208663027052145151285, −9.950475968911032075406519765824, −9.391793020062158127027969773399, −6.79872609013526368934183295556, −4.92265778742368590446764917526, −3.44559721735191343697759776917, 1.40274337689440476140896120740, 3.64949633037211230129792935049, 6.25327438564114307861823250811, 8.083416914576591391106574954306, 8.818965611823172819808589523634, 11.73618955805009330521612354235, 12.54938259832956130150818857332, 13.37828100517438036925718529126, 14.29666371884872318688405901770, 16.63555479085348777532298511229

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