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Properties

Label 2-2100-15.2-c1-0-34
Degree $2$
Conductor $2100$
Sign $-0.639 + 0.768i$
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  + (1.38 − 1.04i)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 + (−1.85 − 4.85i)27-s + 6.65·29-s + 0.677·31-s + (−1.26 − 1.67i)33-s + (−5.16 − 5.16i)37-s + (0.450 − 3.19i)39-s + ⋯
L(s)  = 1  + (0.798 − 0.601i)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 + (−0.357 − 0.934i)27-s + 1.23·29-s + 0.121·31-s + (−0.219 − 0.291i)33-s + (−0.849 − 0.849i)37-s + (0.0721 − 0.512i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.639 + 0.768i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.639 + 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.709568178\)
\(L(\frac12)\) \(\approx\) \(1.709568178\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 + 1.04i)T \)
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

−8.679869874295704085616815330192, −8.212574288387996591055038703549, −7.29788064381197339875092531531, −6.47135240864330419104095745398, −6.00844606553332204191571476561, −4.52761806579064217096657393938, −3.81111486727529130374654126915, −2.79681132005574841706842917434, −1.91761865846680198987108243702, −0.51211555611829742544784072490, 1.71417429684275932830454335850, 2.67390321531137622510953860993, 3.61370552860995262980043915333, 4.43507158543269067926633046770, 5.22483494463727040675792104417, 6.29728548732024540161057066691, 7.12696792965857764291459081129, 8.028643702260588534556745585097, 8.631442890619107422736908678240, 9.474783979018641566925847914977

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