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Properties

Label 2-2100-35.3-c1-0-5
Degree $2$
Conductor $2100$
Sign $-0.869 - 0.493i$
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 + (−1.15 + 2.38i)7-s + (−0.866 + 0.499i)9-s + (−1.36 + 2.36i)11-s + (−0.707 + 0.707i)13-s + (6.50 − 1.74i)17-s + (4.23 + 7.33i)19-s + (−2.59 − 0.500i)21-s + (1.93 − 7.20i)23-s + (−0.707 − 0.707i)27-s + 6.92i·29-s + (−1.73 − i)31-s + (−2.63 − 0.707i)33-s + (−10.8 − 2.89i)37-s + (−0.866 − 0.500i)39-s + ⋯
L(s)  = 1  + (0.149 + 0.557i)3-s + (−0.436 + 0.899i)7-s + (−0.288 + 0.166i)9-s + (−0.411 + 0.713i)11-s + (−0.196 + 0.196i)13-s + (1.57 − 0.422i)17-s + (0.970 + 1.68i)19-s + (−0.566 − 0.109i)21-s + (0.402 − 1.50i)23-s + (−0.136 − 0.136i)27-s + 1.28i·29-s + (−0.311 − 0.179i)31-s + (−0.459 − 0.123i)33-s + (−1.77 − 0.476i)37-s + (−0.138 − 0.0800i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.869 - 0.493i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.869 - 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.250313708\)
\(L(\frac12)\) \(\approx\) \(1.250313708\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.15 - 2.38i)T \)
good11 \( 1 + (1.36 - 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T - 13iT^{2} \)
17 \( 1 + (-6.50 + 1.74i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.23 - 7.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.93 + 7.20i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + (1.73 + i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (10.8 + 2.89i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.19iT - 41T^{2} \)
43 \( 1 + (0.378 + 0.378i)T + 43iT^{2} \)
47 \( 1 + (2.63 - 9.84i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (8.43 - 2.26i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (4.63 - 8.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.42 - 4.86i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.81 - 6.76i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (3.98 + 14.8i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.96 + 1.13i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.34 - 3.34i)T - 83iT^{2} \)
89 \( 1 + (-3.63 - 6.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.50 - 2.50i)T + 97iT^{2} \)
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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

−9.371893021195597005393790951270, −8.855798621225761940568470920072, −7.86302271931072363037325723496, −7.25074849946347463488443252947, −6.08693759298869467785187889846, −5.40832471598858605363957372920, −4.72401031589193785149321426090, −3.47799716979080653263062776663, −2.88119116509988824600101034569, −1.61860235675289328476401485984, 0.43661321582566650251064518383, 1.53518587654110092093451179492, 3.16625780137755999327350384651, 3.38325956234557770621002199820, 4.88537629750441634198934327551, 5.61081951681273896777995623774, 6.54315320690757390459093713159, 7.39247082439324856352465451099, 7.76693940342747100189929814471, 8.704267726108446278977532872681

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