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Properties

Label 2-2100-35.17-c1-0-21
Degree $2$
Conductor $2100$
Sign $0.113 + 0.993i$
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  + (0.965 − 0.258i)3-s + (0.728 + 2.54i)7-s + (0.866 − 0.499i)9-s + (2.14 − 3.71i)11-s + (−3.22 − 3.22i)13-s + (−0.923 − 3.44i)17-s + (−3.13 − 5.42i)19-s + (1.36 + 2.26i)21-s + (−2.79 − 0.749i)23-s + (0.707 − 0.707i)27-s + 2.84i·29-s + (−1.34 − 0.775i)31-s + (1.11 − 4.14i)33-s + (1.47 − 5.50i)37-s + (−3.95 − 2.28i)39-s + ⋯
L(s)  = 1  + (0.557 − 0.149i)3-s + (0.275 + 0.961i)7-s + (0.288 − 0.166i)9-s + (0.646 − 1.12i)11-s + (−0.895 − 0.895i)13-s + (−0.223 − 0.835i)17-s + (−0.719 − 1.24i)19-s + (0.297 + 0.494i)21-s + (−0.583 − 0.156i)23-s + (0.136 − 0.136i)27-s + 0.527i·29-s + (−0.241 − 0.139i)31-s + (0.193 − 0.721i)33-s + (0.242 − 0.905i)37-s + (−0.633 − 0.365i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.113 + 0.993i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.113 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.813621734\)
\(L(\frac12)\) \(\approx\) \(1.813621734\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (-0.728 - 2.54i)T \)
good11 \( 1 + (-2.14 + 3.71i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.22 + 3.22i)T + 13iT^{2} \)
17 \( 1 + (0.923 + 3.44i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.13 + 5.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.79 + 0.749i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.84iT - 29T^{2} \)
31 \( 1 + (1.34 + 0.775i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.47 + 5.50i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 10.7iT - 41T^{2} \)
43 \( 1 + (2.38 - 2.38i)T - 43iT^{2} \)
47 \( 1 + (-8.28 - 2.22i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.34 - 8.76i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.820 - 1.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.94 + 5.16i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.46 - 1.19i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + (-8.97 + 2.40i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.68 + 3.86i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.32 - 3.32i)T + 83iT^{2} \)
89 \( 1 + (8.91 + 15.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.100 + 0.100i)T - 97iT^{2} \)
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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

−9.070901809458567162971428311037, −8.247966916695086363008774068053, −7.46756493167120839225698626218, −6.62203659010162749634925516446, −5.70020170204950522760957548828, −4.99514765836395329117620923862, −3.88996682358320722053812772404, −2.81498148292094860312207403015, −2.23145880140793304053282947355, −0.57924355921620331731766781234, 1.53388167351163490685157863958, 2.25947249144418364102986068675, 3.87407463880491863547805907577, 4.13089738569634465477315084816, 5.05311689431542056103794244029, 6.43343886468736492870833232060, 6.93250634207258243947743799196, 7.85598607840813228119601405856, 8.345350978763330863490004182198, 9.489648063564019941098037181839

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