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Properties

Label 2-2100-35.3-c1-0-6
Degree $2$
Conductor $2100$
Sign $0.284 - 0.958i$
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 + (2.57 + 0.588i)7-s + (−0.866 + 0.499i)9-s + (−0.401 + 0.694i)11-s + (−2.60 + 2.60i)13-s + (0.774 − 0.207i)17-s + (1.40 + 2.43i)19-s + (−0.0989 − 2.64i)21-s + (−2.17 + 8.12i)23-s + (0.707 + 0.707i)27-s + (−5.14 − 2.96i)31-s + (0.774 + 0.207i)33-s + (−1.67 − 0.448i)37-s + (3.18 + 1.83i)39-s + 7.01i·41-s + ⋯
L(s)  = 1  + (−0.149 − 0.557i)3-s + (0.974 + 0.222i)7-s + (−0.288 + 0.166i)9-s + (−0.120 + 0.209i)11-s + (−0.721 + 0.721i)13-s + (0.187 − 0.0503i)17-s + (0.322 + 0.559i)19-s + (−0.0215 − 0.576i)21-s + (−0.453 + 1.69i)23-s + (0.136 + 0.136i)27-s + (−0.923 − 0.533i)31-s + (0.134 + 0.0361i)33-s + (−0.275 − 0.0736i)37-s + (0.510 + 0.294i)39-s + 1.09i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.284 - 0.958i$
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: \((2,\ 2100,\ (\ :1/2),\ 0.284 - 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.306409928\)
\(L(\frac12)\) \(\approx\) \(1.306409928\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.57 - 0.588i)T \)
good11 \( 1 + (0.401 - 0.694i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.60 - 2.60i)T - 13iT^{2} \)
17 \( 1 + (-0.774 + 0.207i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.40 - 2.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.17 - 8.12i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (5.14 + 2.96i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.67 + 0.448i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.01iT - 41T^{2} \)
43 \( 1 + (2.23 + 2.23i)T + 43iT^{2} \)
47 \( 1 + (2.73 - 10.1i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (12.2 - 3.27i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.50 + 6.07i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.78 + 5.07i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.728 - 2.71i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 5.35T + 71T^{2} \)
73 \( 1 + (-2.06 - 7.68i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.85 + 3.38i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.112 - 0.112i)T - 83iT^{2} \)
89 \( 1 + (0.694 + 1.20i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.53 - 3.53i)T + 97iT^{2} \)
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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

−9.376509468383400112140476176892, −8.248976015525175484238082823595, −7.70958102031317630799287739688, −7.10633878646877959091116285580, −6.05288054747186316304154248130, −5.31157593573569253820344648077, −4.55494807618572612776775919007, −3.42790219169870586347030639106, −2.14352307592789403322988541046, −1.43244520675423611755453611416, 0.46626915738195598774621906736, 2.02657665311987756211415336710, 3.10881385892954392122032653807, 4.15257219644671297737245673064, 5.00000453761764958191301337199, 5.47364321713766999350386352149, 6.63506664946690589391407289055, 7.46004910446385556911039324258, 8.279508495913366027859279704718, 8.832858007219727101563154826225

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