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Properties

Label 2-2100-21.17-c1-0-3
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: $\chi_{2100} (101, \cdot )$
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

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

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