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Properties

Label 1-2900-2900.1259-r0-0-0
Degree $1$
Conductor $2900$
Sign $0.769 + 0.638i$
Analytic cond. $13.4675$
Root an. cond. $13.4675$
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  + (−0.587 − 0.809i)3-s + 7-s + (−0.309 + 0.951i)9-s + (0.951 − 0.309i)11-s + (0.309 − 0.951i)13-s + (−0.587 + 0.809i)17-s + (−0.587 + 0.809i)19-s + (−0.587 − 0.809i)21-s + (0.309 + 0.951i)23-s + (0.951 − 0.309i)27-s + (−0.587 + 0.809i)31-s + (−0.809 − 0.587i)33-s + (0.951 + 0.309i)37-s + (−0.951 + 0.309i)39-s + (−0.951 − 0.309i)41-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)3-s + 7-s + (−0.309 + 0.951i)9-s + (0.951 − 0.309i)11-s + (0.309 − 0.951i)13-s + (−0.587 + 0.809i)17-s + (−0.587 + 0.809i)19-s + (−0.587 − 0.809i)21-s + (0.309 + 0.951i)23-s + (0.951 − 0.309i)27-s + (−0.587 + 0.809i)31-s + (−0.809 − 0.587i)33-s + (0.951 + 0.309i)37-s + (−0.951 + 0.309i)39-s + (−0.951 − 0.309i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2900\)    =    \(2^{2} \cdot 5^{2} \cdot 29\)
Sign: $0.769 + 0.638i$
Analytic conductor: \(13.4675\)
Root analytic conductor: \(13.4675\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2900} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2900,\ (0:\ ),\ 0.769 + 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.245145049 + 0.4495162775i\)
\(L(\frac12)\) \(\approx\) \(1.245145049 + 0.4495162775i\)
\(L(1)\) \(\approx\) \(0.9794175354 - 0.08487510429i\)
\(L(1)\) \(\approx\) \(0.9794175354 - 0.08487510429i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + T \)
11 \( 1 + (0.951 - 0.309i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.587 + 0.809i)T \)
19 \( 1 + (-0.587 + 0.809i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (-0.587 + 0.809i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.587 + 0.809i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−18.907444554090105172035402205630, −18.22380024228913180086712078332, −17.583189097994954539919834664024, −16.75671578690632424568261623586, −16.59167461156919123229981711855, −15.34444381569047690084550951394, −15.05672151246399128678242286419, −14.23112390895486815856204810190, −13.601727711179780563550795887038, −12.44732023641550301089742047083, −11.72847265533725662192809434341, −11.23855357531959622260700584463, −10.71790767969725505424976723121, −9.69376374973194259942538126820, −8.99099599907241009836125250264, −8.60008986667737679646597766420, −7.262986961551803437703438072538, −6.67094757313880451137402171315, −5.879532538941210631182677496707, −4.83932427458662329906564106417, −4.45479221976070009198239561626, −3.79833537679245032446553108617, −2.53004486150152024633901988082, −1.646399289417876727057923140292, −0.4701171350343129412457048670, 1.20739628894468737801026280880, 1.497886099297763071314703458653, 2.60414439943469634286003305196, 3.73412980502859335671114178728, 4.55024613165116182727780140234, 5.54483400368678949611565234167, 5.98894962778597985217417414278, 6.858637318031540597573155790982, 7.66206169954154471389253250419, 8.327310307574938733876561555655, 8.88494041414844569062507156697, 10.17197872273215260015171107270, 10.86665600701240868504253765349, 11.40416878288629120973076171185, 12.07954971909833507401957696911, 12.852922791958231657601192258, 13.44222825210254299446049902631, 14.30976829346373886234293150548, 14.861754401801113073389884001306, 15.70847457153281384500731253703, 16.77331524291449447081143154148, 17.1369529325879413655966448530, 17.87213239976910312473149111785, 18.3001716972565626067846005793, 19.240350274496112905916332135069

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