↑

Properties

Label 1-980-980.407-r0-0-0
Degree $1$
Conductor $980$
Sign $0.935 + 0.353i$
Analytic cond. $4.55110$
Root an. cond. $4.55110$
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  + (0.974 − 0.222i)3-s + (0.900 − 0.433i)9-s + (0.900 + 0.433i)11-s + (−0.433 + 0.900i)13-s + (0.781 + 0.623i)17-s + 19-s + (−0.781 + 0.623i)23-s + (0.781 − 0.623i)27-s + (−0.623 + 0.781i)29-s − 31-s + (0.974 + 0.222i)33-s + (0.781 + 0.623i)37-s + (−0.222 + 0.974i)39-s + (−0.222 − 0.974i)41-s + (−0.974 − 0.222i)43-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)3-s + (0.900 − 0.433i)9-s + (0.900 + 0.433i)11-s + (−0.433 + 0.900i)13-s + (0.781 + 0.623i)17-s + 19-s + (−0.781 + 0.623i)23-s + (0.781 − 0.623i)27-s + (−0.623 + 0.781i)29-s − 31-s + (0.974 + 0.222i)33-s + (0.781 + 0.623i)37-s + (−0.222 + 0.974i)39-s + (−0.222 − 0.974i)41-s + (−0.974 − 0.222i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 + 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.935 + 0.353i$
Analytic conductor: \(4.55110\)
Root analytic conductor: \(4.55110\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 980,\ (0:\ ),\ 0.935 + 0.353i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.270809965 + 0.4146634632i\)
\(L(\frac12)\) \(\approx\) \(2.270809965 + 0.4146634632i\)
\(L(1)\) \(\approx\) \(1.570295219 + 0.07483347803i\)
\(L(1)\) \(\approx\) \(1.570295219 + 0.07483347803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.974 - 0.222i)T \)
11 \( 1 + (0.900 + 0.433i)T \)
13 \( 1 + (-0.433 + 0.900i)T \)
17 \( 1 + (0.781 + 0.623i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.781 + 0.623i)T \)
29 \( 1 + (-0.623 + 0.781i)T \)
31 \( 1 - T \)
37 \( 1 + (0.781 + 0.623i)T \)
41 \( 1 + (-0.222 - 0.974i)T \)
43 \( 1 + (-0.974 - 0.222i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (0.781 - 0.623i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (0.623 - 0.781i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (0.433 + 0.900i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.433 - 0.900i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + iT \)
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

−21.74805833024066522617991388879, −20.66060209660265729324452222397, −20.127215957605724461303543290874, −19.52750292856316616287979521351, −18.55685302104595152766089500759, −17.97386187883566835064378213781, −16.658061358309791837868177637876, −16.24009158626917082766975560102, −15.11549814365098659012055090310, −14.57751563937780883850004186978, −13.839228868599897303884643602589, −13.056291489947753496413870437665, −12.1067300271489040296294027279, −11.25539233069542049096362039665, −10.02746027685950645377842283396, −9.63221971476863080332476486762, −8.64963120930251623899627949258, −7.83241188291783552399710129281, −7.16396831838223977300124054151, −5.92667292084713283699616623359, −4.97800642992325890443989572457, −3.8545634478395658522757650180, −3.19118577599403573949971777457, −2.20504358356462793900489568080, −0.97624363601436087131166983553, 1.3957558473467530496899152211, 2.01753900695299011920160323447, 3.366483699414736465603730534342, 3.92086483436542884097028223635, 5.05502339480678777139262681373, 6.27916688285032113856758287737, 7.19860176975935382842342531020, 7.781949694943528120208610344319, 8.8955808112335931047440996797, 9.48963865046933947866112299873, 10.18658443684219413374973365978, 11.559933647722157220403041248553, 12.18867579867393423397662026065, 13.05819028279665187281066864669, 14.02960476815664409815436638823, 14.49351458964147408371742956807, 15.21221284113519877746452604392, 16.25992488643832669491213967796, 16.96580709623182496066135505446, 18.0393308173273068511481359486, 18.695453791901524292958889618329, 19.59285238532354425924880218539, 20.01912110838992773177121203901, 20.86638152096789907132001706210, 21.759429326680708911282630707404

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