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Properties

Label 2-2100-35.12-c1-0-1
Degree $2$
Conductor $2100$
Sign $-0.925 - 0.379i$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)3-s + (1.38 + 2.25i)7-s + (−0.866 − 0.499i)9-s + (−2.42 − 4.19i)11-s + (−2.27 − 2.27i)13-s + (2.14 + 0.574i)17-s + (0.107 − 0.185i)19-s + (−2.53 + 0.758i)21-s + (1.78 + 6.64i)23-s + (0.707 − 0.707i)27-s + 9.28i·29-s + (−7.78 + 4.49i)31-s + (4.68 − 1.25i)33-s + (−9.78 + 2.62i)37-s + (2.78 − 1.61i)39-s + ⋯
L(s)  = 1  + (−0.149 + 0.557i)3-s + (0.524 + 0.851i)7-s + (−0.288 − 0.166i)9-s + (−0.730 − 1.26i)11-s + (−0.631 − 0.631i)13-s + (0.520 + 0.139i)17-s + (0.0246 − 0.0426i)19-s + (−0.553 + 0.165i)21-s + (0.371 + 1.38i)23-s + (0.136 − 0.136i)27-s + 1.72i·29-s + (−1.39 + 0.807i)31-s + (0.815 − 0.218i)33-s + (−1.60 + 0.431i)37-s + (0.446 − 0.257i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.925 - 0.379i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.925 - 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7595149929\)
\(L(\frac12)\) \(\approx\) \(0.7595149929\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (-1.38 - 2.25i)T \)
good11 \( 1 + (2.42 + 4.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.27 + 2.27i)T + 13iT^{2} \)
17 \( 1 + (-2.14 - 0.574i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.107 + 0.185i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.78 - 6.64i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 9.28iT - 29T^{2} \)
31 \( 1 + (7.78 - 4.49i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.78 - 2.62i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.32iT - 41T^{2} \)
43 \( 1 + (-5.01 + 5.01i)T - 43iT^{2} \)
47 \( 1 + (-2.50 - 9.36i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.93 + 0.785i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.68 - 4.64i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (13.2 + 7.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.33 + 4.97i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (0.0391 - 0.146i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-0.0599 - 0.0346i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.467 - 0.467i)T + 83iT^{2} \)
89 \( 1 + (-0.997 + 1.72i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.51 + 7.51i)T - 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.193973468550983316581085305563, −8.869013044647463996986894093781, −7.939958001274361782248725506280, −7.28280682584518045104289904769, −5.96490578463835372720903677421, −5.35728287053905649547774518518, −4.96355548922238915631420640455, −3.42276860647128298955378512552, −2.98449340729774668744288128268, −1.51776700769685611037143995795, 0.26269404479695057238954641906, 1.76361980158757031652614092021, 2.52074018561502579610876675639, 4.01079692226764324749888961393, 4.69572797290338205938904855733, 5.51620565090508725254397172740, 6.61477799267588664337011670672, 7.43795209949011611543110867843, 7.60406344583778488694305417322, 8.678284368894141252312551336992

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