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Properties

Label 2-2100-21.5-c1-0-38
Degree $2$
Conductor $2100$
Sign $0.153 + 0.988i$
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  + (0.772 + 1.55i)3-s + (−2.60 − 0.477i)7-s + (−1.80 + 2.39i)9-s + (−1.34 + 0.773i)11-s − 4.18i·13-s + (2.55 + 4.42i)17-s + (−4.62 − 2.67i)19-s + (−1.26 − 4.40i)21-s + (−3.15 − 1.82i)23-s + (−5.10 − 0.954i)27-s − 9.79i·29-s + (6.79 − 3.92i)31-s + (−2.23 − 1.48i)33-s + (−1.71 + 2.96i)37-s + (6.48 − 3.23i)39-s + ⋯
L(s)  = 1  + (0.445 + 0.895i)3-s + (−0.983 − 0.180i)7-s + (−0.602 + 0.798i)9-s + (−0.404 + 0.233i)11-s − 1.16i·13-s + (0.619 + 1.07i)17-s + (−1.06 − 0.613i)19-s + (−0.276 − 0.960i)21-s + (−0.658 − 0.380i)23-s + (−0.982 − 0.183i)27-s − 1.81i·29-s + (1.22 − 0.705i)31-s + (−0.389 − 0.257i)33-s + (−0.281 + 0.488i)37-s + (1.03 − 0.517i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.153 + 0.988i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.153 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7665022273\)
\(L(\frac12)\) \(\approx\) \(0.7665022273\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.772 - 1.55i)T \)
5 \( 1 \)
7 \( 1 + (2.60 + 0.477i)T \)
good11 \( 1 + (1.34 - 0.773i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.18iT - 13T^{2} \)
17 \( 1 + (-2.55 - 4.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.62 + 2.67i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.15 + 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.79iT - 29T^{2} \)
31 \( 1 + (-6.79 + 3.92i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.71 - 2.96i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.82T + 41T^{2} \)
43 \( 1 + 3.79T + 43T^{2} \)
47 \( 1 + (-1.24 + 2.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.684 + 0.395i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.73 + 4.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.76 + 3.90i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.58 + 9.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 7.97iT - 71T^{2} \)
73 \( 1 + (10.7 - 6.19i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.12 + 1.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.26T + 83T^{2} \)
89 \( 1 + (-7.65 + 13.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.9iT - 97T^{2} \)
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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

−8.997460835938991167509417454621, −8.101037421969735684389862779951, −7.73465467097340782158934260360, −6.28794235189874168028524212689, −5.89390610916724576519260185240, −4.69072801653874477415202986521, −3.99342391413461686591317439219, −3.08463983055600934680169589214, −2.30784443533039174080874300311, −0.25123152195112546172472890482, 1.32733958622931573803337452517, 2.51902986011308728975727348170, 3.23691707133303790462541300426, 4.26681341482453501415441135573, 5.55024590073685396317241313025, 6.28469121161835091742434503798, 6.98009025911272545979277241652, 7.59759333555824261793573008007, 8.631854385532377203674547326636, 9.071011488641557149679342249150

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