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Properties

Label 2-2100-15.8-c1-0-8
Degree $2$
Conductor $2100$
Sign $-0.401 - 0.915i$
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  + (1.04 + 1.38i)3-s + (−0.707 + 0.707i)7-s + (−0.829 + 2.88i)9-s − 1.21i·11-s + (1.31 + 1.31i)13-s + (4.79 + 4.79i)17-s + 3.66i·19-s + (−1.71 − 0.241i)21-s + (6.15 − 6.15i)23-s + (−4.85 + 1.85i)27-s − 6.65·29-s + 0.677·31-s + (1.67 − 1.26i)33-s + (−5.16 + 5.16i)37-s + (−0.450 + 3.19i)39-s + ⋯
L(s)  = 1  + (0.601 + 0.798i)3-s + (−0.267 + 0.267i)7-s + (−0.276 + 0.961i)9-s − 0.365i·11-s + (0.365 + 0.365i)13-s + (1.16 + 1.16i)17-s + 0.841i·19-s + (−0.374 − 0.0527i)21-s + (1.28 − 1.28i)23-s + (−0.934 + 0.357i)27-s − 1.23·29-s + 0.121·31-s + (0.291 − 0.219i)33-s + (−0.849 + 0.849i)37-s + (−0.0721 + 0.512i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.401 - 0.915i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.401 - 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.954911209\)
\(L(\frac12)\) \(\approx\) \(1.954911209\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 - 1.38i)T \)
5 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 + 1.21iT - 11T^{2} \)
13 \( 1 + (-1.31 - 1.31i)T + 13iT^{2} \)
17 \( 1 + (-4.79 - 4.79i)T + 17iT^{2} \)
19 \( 1 - 3.66iT - 19T^{2} \)
23 \( 1 + (-6.15 + 6.15i)T - 23iT^{2} \)
29 \( 1 + 6.65T + 29T^{2} \)
31 \( 1 - 0.677T + 31T^{2} \)
37 \( 1 + (5.16 - 5.16i)T - 37iT^{2} \)
41 \( 1 + 4.65iT - 41T^{2} \)
43 \( 1 + (-2.35 - 2.35i)T + 43iT^{2} \)
47 \( 1 + (-1.91 - 1.91i)T + 47iT^{2} \)
53 \( 1 + (8.64 - 8.64i)T - 53iT^{2} \)
59 \( 1 - 6.62T + 59T^{2} \)
61 \( 1 + 7.07T + 61T^{2} \)
67 \( 1 + (4.94 - 4.94i)T - 67iT^{2} \)
71 \( 1 - 1.87iT - 71T^{2} \)
73 \( 1 + (-7.93 - 7.93i)T + 73iT^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + (9.69 - 9.69i)T - 83iT^{2} \)
89 \( 1 - 2.84T + 89T^{2} \)
97 \( 1 + (3.90 - 3.90i)T - 97iT^{2} \)
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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

−9.316894178434075577273636425860, −8.571130487804803972412966837856, −8.102015544081032573428884689897, −7.09143808861047512794115829261, −6.01671602985404501889988458736, −5.40257866674918709660691956521, −4.30812613632452565192910794544, −3.57087193217341961730600145053, −2.78956694306445879276036737306, −1.51028254207065357327198632315, 0.65540727636689629417756592842, 1.80100149646361213434145874533, 3.05445053073158561087723905044, 3.53109402265507653854473481173, 4.93541168416180511279564783888, 5.72527359003930453720638756270, 6.76490521936703527348325700034, 7.41710689629789248695966850420, 7.80879631359206205059778940061, 9.087565868618663965217476495346

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