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Properties

Label 2-2100-15.8-c1-0-24
Degree $2$
Conductor $2100$
Sign $0.303 + 0.952i$
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  + (−0.0412 + 1.73i)3-s + (−0.707 + 0.707i)7-s + (−2.99 − 0.142i)9-s − 1.76i·11-s + (0.719 + 0.719i)13-s + (−1.55 − 1.55i)17-s − 5.12i·19-s + (−1.19 − 1.25i)21-s + (−1.47 + 1.47i)23-s + (0.371 − 5.18i)27-s − 7.63·29-s − 0.104·31-s + (3.05 + 0.0728i)33-s + (0.0126 − 0.0126i)37-s + (−1.27 + 1.21i)39-s + ⋯
L(s)  = 1  + (−0.0238 + 0.999i)3-s + (−0.267 + 0.267i)7-s + (−0.998 − 0.0476i)9-s − 0.532i·11-s + (0.199 + 0.199i)13-s + (−0.376 − 0.376i)17-s − 1.17i·19-s + (−0.260 − 0.273i)21-s + (−0.306 + 0.306i)23-s + (0.0714 − 0.997i)27-s − 1.41·29-s − 0.0187·31-s + (0.532 + 0.0126i)33-s + (0.00207 − 0.00207i)37-s + (−0.204 + 0.194i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.303 + 0.952i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.303 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7400526132\)
\(L(\frac12)\) \(\approx\) \(0.7400526132\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0412 - 1.73i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + 1.76iT - 11T^{2} \)
13 \( 1 + (-0.719 - 0.719i)T + 13iT^{2} \)
17 \( 1 + (1.55 + 1.55i)T + 17iT^{2} \)
19 \( 1 + 5.12iT - 19T^{2} \)
23 \( 1 + (1.47 - 1.47i)T - 23iT^{2} \)
29 \( 1 + 7.63T + 29T^{2} \)
31 \( 1 + 0.104T + 31T^{2} \)
37 \( 1 + (-0.0126 + 0.0126i)T - 37iT^{2} \)
41 \( 1 + 9.35iT - 41T^{2} \)
43 \( 1 + (-2.66 - 2.66i)T + 43iT^{2} \)
47 \( 1 + (3.98 + 3.98i)T + 47iT^{2} \)
53 \( 1 + (-5.44 + 5.44i)T - 53iT^{2} \)
59 \( 1 + 5.32T + 59T^{2} \)
61 \( 1 + 4.82T + 61T^{2} \)
67 \( 1 + (-3.01 + 3.01i)T - 67iT^{2} \)
71 \( 1 + 5.20iT - 71T^{2} \)
73 \( 1 + (-7.10 - 7.10i)T + 73iT^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + (-3.76 + 3.76i)T - 83iT^{2} \)
89 \( 1 + 2.98T + 89T^{2} \)
97 \( 1 + (6.41 - 6.41i)T - 97iT^{2} \)
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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

−9.126406717919458012484077335355, −8.431972133086650326420499522389, −7.41274310464912480023881656830, −6.46654101991078410323234350069, −5.63629552112229625902883753466, −4.95282592781951789639168075429, −3.97597983461617508874050224606, −3.22558067070104083386793670394, −2.19596437024145144271352604240, −0.26255818239223077816644957958, 1.30224728361466434565305779000, 2.24750343798655599506148785008, 3.37522510598477408410635900241, 4.32766815946285435682169287462, 5.54417885230870426417887800091, 6.18000201144246650865566738363, 6.94717732006341497544973046671, 7.72794903936003184874009458933, 8.256155024987863338892161336164, 9.206550939297261120390620729587

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