Properties

Label 2-5e2-25.3-c2-0-2
Degree $2$
Conductor $25$
Sign $0.680 + 0.732i$
Analytic cond. $0.681200$
Root an. cond. $0.825348$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.395 − 0.776i)2-s + (−0.296 − 1.87i)3-s + (1.90 − 2.62i)4-s + (1.22 + 4.84i)5-s + (−1.33 + 0.970i)6-s + (−5.60 + 5.60i)7-s + (−6.23 − 0.986i)8-s + (5.14 − 1.67i)9-s + (3.27 − 2.87i)10-s + (−4.46 + 13.7i)11-s + (−5.47 − 2.78i)12-s + (5.52 − 10.8i)13-s + (6.57 + 2.13i)14-s + (8.70 − 3.73i)15-s + (−2.30 − 7.10i)16-s + (0.147 − 0.932i)17-s + ⋯
L(s)  = 1  + (−0.197 − 0.388i)2-s + (−0.0988 − 0.623i)3-s + (0.476 − 0.655i)4-s + (0.245 + 0.969i)5-s + (−0.222 + 0.161i)6-s + (−0.801 + 0.801i)7-s + (−0.778 − 0.123i)8-s + (0.571 − 0.185i)9-s + (0.327 − 0.287i)10-s + (−0.406 + 1.25i)11-s + (−0.456 − 0.232i)12-s + (0.425 − 0.834i)13-s + (0.469 + 0.152i)14-s + (0.580 − 0.249i)15-s + (−0.144 − 0.443i)16-s + (0.00869 − 0.0548i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.680 + 0.732i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25\)    =    \(5^{2}\)
Sign: $0.680 + 0.732i$
Analytic conductor: \(0.681200\)
Root analytic conductor: \(0.825348\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{25} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 25,\ (\ :1),\ 0.680 + 0.732i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.808417 - 0.352644i\)
\(L(\frac12)\) \(\approx\) \(0.808417 - 0.352644i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.22 - 4.84i)T \)
good2 \( 1 + (0.395 + 0.776i)T + (-2.35 + 3.23i)T^{2} \)
3 \( 1 + (0.296 + 1.87i)T + (-8.55 + 2.78i)T^{2} \)
7 \( 1 + (5.60 - 5.60i)T - 49iT^{2} \)
11 \( 1 + (4.46 - 13.7i)T + (-97.8 - 71.1i)T^{2} \)
13 \( 1 + (-5.52 + 10.8i)T + (-99.3 - 136. i)T^{2} \)
17 \( 1 + (-0.147 + 0.932i)T + (-274. - 89.3i)T^{2} \)
19 \( 1 + (11.5 + 15.8i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + (4.69 - 2.39i)T + (310. - 427. i)T^{2} \)
29 \( 1 + (7.87 - 10.8i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-12.1 + 8.83i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-57.7 - 29.4i)T + (804. + 1.10e3i)T^{2} \)
41 \( 1 + (16.7 + 51.6i)T + (-1.35e3 + 988. i)T^{2} \)
43 \( 1 + (-3.47 - 3.47i)T + 1.84e3iT^{2} \)
47 \( 1 + (-65.5 + 10.3i)T + (2.10e3 - 682. i)T^{2} \)
53 \( 1 + (6.61 + 41.7i)T + (-2.67e3 + 868. i)T^{2} \)
59 \( 1 + (35.6 - 11.5i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (32.4 - 99.9i)T + (-3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + (7.05 - 44.5i)T + (-4.26e3 - 1.38e3i)T^{2} \)
71 \( 1 + (-36.5 - 26.5i)T + (1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-47.8 + 24.3i)T + (3.13e3 - 4.31e3i)T^{2} \)
79 \( 1 + (-70.5 + 97.1i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (60.5 + 9.59i)T + (6.55e3 + 2.12e3i)T^{2} \)
89 \( 1 + (-48.8 - 15.8i)T + (6.40e3 + 4.65e3i)T^{2} \)
97 \( 1 + (36.3 - 5.75i)T + (8.94e3 - 2.90e3i)T^{2} \)
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

−17.82396037574020397660836976814, −15.53696694371917354545787734367, −15.09628056458961582926411190645, −13.19522294035880671376233948886, −12.09197200985046640482649261615, −10.57066273174379809289595777296, −9.590891891808325520972965546862, −7.14573973507758729723259673492, −6.04647232390251100700420531164, −2.46728936999697706176841423502, 3.97061759697752838619753611061, 6.22477370456599607984679984206, 8.024756812985741207509298017224, 9.438343075774844698712396795800, 10.93357521905109734423185663042, 12.59350450130060521807649699863, 13.64883627917978882192818148673, 15.69295874375335105614944072169, 16.53999584329451062955091405039, 16.79398966767949417503821063580

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