↑

Properties

Label 1-4025-4025.1756-r1-0-0
Degree $1$
Conductor $4025$
Sign $-0.156 + 0.987i$
Analytic cond. $432.546$
Root an. cond. $432.546$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.941 − 0.336i)2-s + (−0.516 − 0.856i)3-s + (0.774 − 0.633i)4-s + (−0.774 − 0.633i)6-s + (0.516 − 0.856i)8-s + (−0.466 + 0.884i)9-s + (0.610 − 0.791i)11-s + (−0.941 − 0.336i)12-s + (0.870 − 0.491i)13-s + (0.198 − 0.980i)16-s + (−0.774 − 0.633i)17-s + (−0.142 + 0.989i)18-s + (0.254 − 0.967i)19-s + (0.309 − 0.951i)22-s − 24-s + ⋯
L(s)  = 1  + (0.941 − 0.336i)2-s + (−0.516 − 0.856i)3-s + (0.774 − 0.633i)4-s + (−0.774 − 0.633i)6-s + (0.516 − 0.856i)8-s + (−0.466 + 0.884i)9-s + (0.610 − 0.791i)11-s + (−0.941 − 0.336i)12-s + (0.870 − 0.491i)13-s + (0.198 − 0.980i)16-s + (−0.774 − 0.633i)17-s + (−0.142 + 0.989i)18-s + (0.254 − 0.967i)19-s + (0.309 − 0.951i)22-s − 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4025\)    =    \(5^{2} \cdot 7 \cdot 23\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(432.546\)
Root analytic conductor: \(432.546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4025} (1756, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4025,\ (1:\ ),\ -0.156 + 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.9902413296 - 1.159011047i\)
\(L(\frac12)\) \(\approx\) \(-0.9902413296 - 1.159011047i\)
\(L(1)\) \(\approx\) \(1.111684473 - 0.9691144858i\)
\(L(1)\) \(\approx\) \(1.111684473 - 0.9691144858i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.941 - 0.336i)T \)
3 \( 1 + (-0.516 - 0.856i)T \)
11 \( 1 + (0.610 - 0.791i)T \)
13 \( 1 + (0.870 - 0.491i)T \)
17 \( 1 + (-0.774 - 0.633i)T \)
19 \( 1 + (0.254 - 0.967i)T \)
29 \( 1 + (-0.254 - 0.967i)T \)
31 \( 1 + (-0.516 + 0.856i)T \)
37 \( 1 + (-0.466 + 0.884i)T \)
41 \( 1 + (-0.696 + 0.717i)T \)
43 \( 1 + (-0.654 + 0.755i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (0.993 - 0.113i)T \)
59 \( 1 + (0.870 - 0.491i)T \)
61 \( 1 + (-0.974 - 0.226i)T \)
67 \( 1 + (-0.0285 - 0.999i)T \)
71 \( 1 + (-0.564 + 0.825i)T \)
73 \( 1 + (0.998 - 0.0570i)T \)
79 \( 1 + (-0.736 - 0.676i)T \)
83 \( 1 + (-0.897 + 0.441i)T \)
89 \( 1 + (-0.0855 + 0.996i)T \)
97 \( 1 + (-0.897 - 0.441i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.6661262060278943704054419399, −17.870810322596217161723403014932, −17.1625704800215542083135670590, −16.57612567333449233675397815639, −16.09190956152520939310866978072, −15.23124088869736408093806466264, −14.899958431027846099492183585177, −14.15862711001687748484242843101, −13.40793812395570108888992206433, −12.55740490127284202909499625348, −11.96972236470620837330587787972, −11.36071360918103660364499920399, −10.660104379807773892399421330531, −10.017312287034789801744447296532, −8.940632697057321743632938183166, −8.516072871232579385190995679531, −7.24271045981099796662908028404, −6.740437468346061678335572337842, −5.90181354444064695968620926486, −5.41720120661808067529187263810, −4.49178715263655701786331254134, −3.84401181499338216666331584834, −3.54438460959543523764016163782, −2.183564203731721930097682781076, −1.44667951802171373291893543832, 0.160419495226703225983487624313, 1.02358178427336048569434311208, 1.66065010720970759974923694657, 2.7295913286934946009345251124, 3.26813345608592307502421520667, 4.32160590110195349557286159451, 5.10473423707718046472938051726, 5.74947209056762178209524343697, 6.57467509579657439146395020691, 6.82082452234708464221684122838, 7.893159351657238273838522728629, 8.65028242974955805098875825939, 9.60474873244711367679194855913, 10.6225729171853522309643943567, 11.29430560744161390559878616883, 11.52191355581509017701002980148, 12.35098925135633730331979156160, 13.226593443126771838117973660671, 13.49419573001769468459164323905, 14.07101249658748535156821758476, 15.00874076332602042614368779377, 15.770058657886549725598411315575, 16.34112692193747443433431104486, 17.10539323101907166139118943601, 17.96201456503150987526171986641

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