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Properties

Label 1-3920-3920.29-r0-0-0
Degree $1$
Conductor $3920$
Sign $0.426 + 0.904i$
Analytic cond. $18.2044$
Root an. cond. $18.2044$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.433 + 0.900i)3-s + (−0.623 + 0.781i)9-s + (0.781 − 0.623i)11-s + (0.781 − 0.623i)13-s + (0.222 + 0.974i)17-s + i·19-s + (−0.222 + 0.974i)23-s + (−0.974 − 0.222i)27-s + (0.974 − 0.222i)29-s + 31-s + (0.900 + 0.433i)33-s + (0.974 − 0.222i)37-s + (0.900 + 0.433i)39-s + (0.900 − 0.433i)41-s + (0.433 − 0.900i)43-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)3-s + (−0.623 + 0.781i)9-s + (0.781 − 0.623i)11-s + (0.781 − 0.623i)13-s + (0.222 + 0.974i)17-s + i·19-s + (−0.222 + 0.974i)23-s + (−0.974 − 0.222i)27-s + (0.974 − 0.222i)29-s + 31-s + (0.900 + 0.433i)33-s + (0.974 − 0.222i)37-s + (0.900 + 0.433i)39-s + (0.900 − 0.433i)41-s + (0.433 − 0.900i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.426 + 0.904i$
Analytic conductor: \(18.2044\)
Root analytic conductor: \(18.2044\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3920,\ (0:\ ),\ 0.426 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.015968609 + 1.278013192i\)
\(L(\frac12)\) \(\approx\) \(2.015968609 + 1.278013192i\)
\(L(1)\) \(\approx\) \(1.311695625 + 0.4436991784i\)
\(L(1)\) \(\approx\) \(1.311695625 + 0.4436991784i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−18.36734394088374991538936182302, −17.82419490590137946219995345680, −17.26172860073853190480521682316, −16.30917685296192719482874099689, −15.72740204010063394526154444465, −14.75685926400136795217500545870, −14.19531297576734803561191574497, −13.74455053287159306740383177322, −12.84282252089790875181047413770, −12.37140746258157580546678799805, −11.46494042481773763827179927556, −11.16902063684460464008685795800, −9.76067830256198683260683649711, −9.39018083273982473803608402654, −8.51362050977035607424111021481, −7.989473501449649773530853781273, −6.93223820545909073794523697836, −6.665138954375955016924156131275, −5.91937698196963854727219139363, −4.67667255710251232862162729147, −4.17273120332991173608746308757, −2.94356461546666559808621283162, −2.53583732887360006893101453672, −1.41433767080411694670668197422, −0.82112526691899333495130286215, 0.94206296081488950487483646516, 1.901985347944260790567838120040, 2.97206909867345708970152388917, 3.69984262892756029765223750370, 4.06310656585153252788196866897, 5.15213089775433320858302653139, 5.92923680109457646275512147887, 6.39617575218438371651415615204, 7.84673328279620049821787620363, 8.142816834886946971298711077478, 8.933631218537112650981725116649, 9.6213125287025326473383314604, 10.37336025003233875201741182474, 10.87406975127611035428307851773, 11.67866136194051178921486408243, 12.40856412485528762240541486543, 13.41584281574823634495875063487, 13.93042785177762057779094367453, 14.58779769047762432631197273587, 15.28737646334937494086296115868, 15.875963177232894448734187466793, 16.52659824233272718525668086053, 17.16712810036326822511919990630, 17.8444117975821763237363009250, 18.81773203584290491951794851531

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