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Properties

Label 2-2100-35.3-c1-0-4
Degree $2$
Conductor $2100$
Sign $0.0449 - 0.998i$
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 + (−2.20 + 1.45i)7-s + (−0.866 + 0.499i)9-s + (0.803 − 1.39i)11-s + (−0.275 + 0.275i)13-s + (0.633 − 0.169i)17-s + (−1.05 − 1.83i)19-s + (1.97 + 1.75i)21-s + (0.622 − 2.32i)23-s + (0.707 + 0.707i)27-s + 7.02i·29-s + (−2.35 − 1.36i)31-s + (−1.55 − 0.415i)33-s + (−2.43 − 0.651i)37-s + (0.337 + 0.194i)39-s + ⋯
L(s)  = 1  + (−0.149 − 0.557i)3-s + (−0.834 + 0.551i)7-s + (−0.288 + 0.166i)9-s + (0.242 − 0.419i)11-s + (−0.0764 + 0.0764i)13-s + (0.153 − 0.0411i)17-s + (−0.242 − 0.420i)19-s + (0.432 + 0.382i)21-s + (0.129 − 0.484i)23-s + (0.136 + 0.136i)27-s + 1.30i·29-s + (−0.423 − 0.244i)31-s + (−0.270 − 0.0723i)33-s + (−0.399 − 0.107i)37-s + (0.0540 + 0.0311i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0449 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0449 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.0449 - 0.998i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.0449 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7885848514\)
\(L(\frac12)\) \(\approx\) \(0.7885848514\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 - 1.45i)T \)
good11 \( 1 + (-0.803 + 1.39i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.275 - 0.275i)T - 13iT^{2} \)
17 \( 1 + (-0.633 + 0.169i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.05 + 1.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.622 + 2.32i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.02iT - 29T^{2} \)
31 \( 1 + (2.35 + 1.36i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.43 + 0.651i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 9.88iT - 41T^{2} \)
43 \( 1 + (-2.73 - 2.73i)T + 43iT^{2} \)
47 \( 1 + (1.44 - 5.40i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.439 - 0.117i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.85 - 4.95i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.0 - 6.40i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.94 - 14.7i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + (-3.70 - 13.8i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.64 + 4.41i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.09 + 5.09i)T - 83iT^{2} \)
89 \( 1 + (-4.64 - 8.04i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.40 + 5.40i)T + 97iT^{2} \)
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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

−9.110750487850111800636079857057, −8.662977848697008035961190596714, −7.66104576194452651590602930489, −6.85554453794543682354763075671, −6.21892631542004197849816365729, −5.51095039469939763421425740508, −4.48034689842044974700463235190, −3.27825550494305181140741360330, −2.55778241772491836249162376387, −1.21564192337565375519885866451, 0.29988710280505966330714369660, 1.94908929502339580139024347548, 3.28887019687082952429306432393, 3.90443308774124632404667409501, 4.81448405546733490604330724560, 5.77692813485315225164169100489, 6.51529557205430710700941419485, 7.33463375133512712344766717609, 8.112890974921260820915841961752, 9.244074509591518464819333538533

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