↑

Properties

Label 2-725-145.4-c1-0-40
Degree $2$
Conductor $725$
Sign $-0.666 + 0.745i$
Analytic cond. $5.78915$
Root an. cond. $2.40606$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.461 − 2.02i)2-s + (2.96 − 1.42i)3-s + (−2.06 − 0.995i)4-s + (−1.51 − 6.64i)6-s + (1.85 + 3.84i)7-s + (−0.381 + 0.478i)8-s + (4.87 − 6.11i)9-s + (−2.20 + 1.75i)11-s − 7.54·12-s + (1.18 − 0.946i)13-s + (8.62 − 1.96i)14-s + (−2.07 − 2.59i)16-s − 3.45·17-s + (−10.0 − 12.6i)18-s + (−0.384 + 0.798i)19-s + ⋯
L(s)  = 1  + (0.326 − 1.42i)2-s + (1.71 − 0.823i)3-s + (−1.03 − 0.497i)4-s + (−0.619 − 2.71i)6-s + (0.699 + 1.45i)7-s + (−0.134 + 0.169i)8-s + (1.62 − 2.03i)9-s + (−0.663 + 0.529i)11-s − 2.17·12-s + (0.329 − 0.262i)13-s + (2.30 − 0.526i)14-s + (−0.517 − 0.649i)16-s − 0.837·17-s + (−2.38 − 2.98i)18-s + (−0.0881 + 0.183i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.666 + 0.745i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.666 + 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31638 - 2.94367i\)
\(L(\frac12)\) \(\approx\) \(1.31638 - 2.94367i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 + (0.716 - 5.33i)T \)
good2 \( 1 + (-0.461 + 2.02i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (-2.96 + 1.42i)T + (1.87 - 2.34i)T^{2} \)
7 \( 1 + (-1.85 - 3.84i)T + (-4.36 + 5.47i)T^{2} \)
11 \( 1 + (2.20 - 1.75i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (-1.18 + 0.946i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + 3.45T + 17T^{2} \)
19 \( 1 + (0.384 - 0.798i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (2.55 - 0.583i)T + (20.7 - 9.97i)T^{2} \)
31 \( 1 + (1.39 + 0.317i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (2.94 - 3.69i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + 6.26iT - 41T^{2} \)
43 \( 1 + (-0.0601 - 0.263i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-2.05 - 2.57i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (7.53 + 1.72i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 - 4.74T + 59T^{2} \)
61 \( 1 + (-1.18 - 2.45i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (-8.60 - 6.85i)T + (14.9 + 65.3i)T^{2} \)
71 \( 1 + (1.09 + 1.36i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.489 - 2.14i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (-13.2 - 10.5i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (-3.14 + 6.52i)T + (-51.7 - 64.8i)T^{2} \)
89 \( 1 + (4.29 + 0.980i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (10.7 + 5.17i)T + (60.4 + 75.8i)T^{2} \)
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

−10.02472939768899708190882480583, −9.167612689144919554460451428699, −8.587478555812483979010340807946, −7.81603494211596327591994683484, −6.77499953008506837669916974177, −5.26162432981979685267417809726, −4.04836024752350113520693766264, −2.98582975003492200801361796782, −2.26113360684613654473947407777, −1.64398698421182940682009418300, 2.12199382851712567158684557456, 3.66904033688696580809211373355, 4.33588343287430464466369946263, 5.07032497973777836952399291018, 6.55414707204286369498701373968, 7.54873788961936257317448721327, 7.989792498283204313559704777199, 8.620255667847951714808302787653, 9.560314939996470095864480751120, 10.59325517320791454027393780989

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