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Properties

Label 1-2624-2624.413-r1-0-0
Degree $1$
Conductor $2624$
Sign $-0.574 + 0.818i$
Analytic cond. $281.987$
Root an. cond. $281.987$
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  + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)5-s − 7-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)11-s + (0.923 − 0.382i)13-s + (−0.707 − 0.707i)15-s + (−0.707 − 0.707i)17-s + (0.382 + 0.923i)19-s + (0.382 + 0.923i)21-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (0.923 + 0.382i)27-s + (0.382 + 0.923i)29-s + 31-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)5-s − 7-s + (−0.707 + 0.707i)9-s + (−0.923 − 0.382i)11-s + (0.923 − 0.382i)13-s + (−0.707 − 0.707i)15-s + (−0.707 − 0.707i)17-s + (0.382 + 0.923i)19-s + (0.382 + 0.923i)21-s + (0.707 + 0.707i)23-s + (0.707 − 0.707i)25-s + (0.923 + 0.382i)27-s + (0.382 + 0.923i)29-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2624\)    =    \(2^{6} \cdot 41\)
Sign: $-0.574 + 0.818i$
Analytic conductor: \(281.987\)
Root analytic conductor: \(281.987\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2624} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2624,\ (1:\ ),\ -0.574 + 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1608949165 - 0.3095428949i\)
\(L(\frac12)\) \(\approx\) \(-0.1608949165 - 0.3095428949i\)
\(L(1)\) \(\approx\) \(0.7920585483 - 0.3477011931i\)
\(L(1)\) \(\approx\) \(0.7920585483 - 0.3477011931i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.382 - 0.923i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 + (-0.707 - 0.707i)T \)
19 \( 1 + (0.382 + 0.923i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 + T \)
37 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (-0.382 - 0.923i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.923 - 0.382i)T \)
59 \( 1 + (-0.382 - 0.923i)T \)
61 \( 1 + (0.382 - 0.923i)T \)
67 \( 1 + (-0.923 + 0.382i)T \)
71 \( 1 - T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.382 - 0.923i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−19.65202834431604778198765690174, −18.90457445930436341366202719159, −17.94297012275526221557983672490, −17.63512734549614208628583592574, −16.70531844084327519888834899923, −16.092181496096517246861951424734, −15.436843360783727620559199068487, −14.89795259627399302625113422425, −13.825097323967075919476766795064, −13.26789144145914937013681611071, −12.64054908344384780360328442170, −11.524947484523323465978126494225, −10.72627950758585123968809576865, −10.3890385980814541939302672925, −9.47281071580694135923222799103, −9.13243836443209959007587267704, −8.12302710068772643167530646151, −6.77761510842341593554025450566, −6.35930558800177161236510259825, −5.685323213686342276520113956082, −4.75946973821394475210902728043, −4.054645923180375052572995978038, −2.899804841683225267589297528634, −2.57210148236006962661058790923, −1.1023768148960481792622941977, 0.06936264560893419904236922751, 0.944222219860910803285127982342, 1.75756847732443082310726615662, 2.80269915761558761719036846029, 3.33589555861745158608006474391, 4.94260344697850972599915775363, 5.4320402213982309512639776709, 6.303892193926465965951396992590, 6.649202837226324711723522921558, 7.73254979047506269188872280714, 8.47267797135693530699240311616, 9.21618348373652947324069253409, 10.130791512695190416345777506085, 10.71314211511709531430767524671, 11.63037248418625571160449765244, 12.45193932706857474869437064897, 13.176502138307293690530226549216, 13.4617960425870315924655098268, 14.020395869666147966786787883202, 15.30050504050753358299566837057, 16.19233287430817362371427140664, 16.47814555996548116343204894315, 17.50574864053003736715456974884, 17.9778013862763223876535178550, 18.69896174288722670201014609518

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