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Properties

Label 2-2100-21.17-c1-0-2
Degree $2$
Conductor $2100$
Sign $-0.998 - 0.0549i$
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  + (0.0597 + 1.73i)3-s + (0.567 − 2.58i)7-s + (−2.99 + 0.206i)9-s + (−0.793 − 0.457i)11-s + 4.31i·13-s + (0.268 − 0.465i)17-s + (1.12 − 0.651i)19-s + (4.50 + 0.827i)21-s + (−6.91 + 3.99i)23-s + (−0.537 − 5.16i)27-s + 3.46i·29-s + (−5.56 − 3.21i)31-s + (0.745 − 1.40i)33-s + (2.52 + 4.36i)37-s + (−7.46 + 0.257i)39-s + ⋯
L(s)  = 1  + (0.0345 + 0.999i)3-s + (0.214 − 0.976i)7-s + (−0.997 + 0.0689i)9-s + (−0.239 − 0.138i)11-s + 1.19i·13-s + (0.0651 − 0.112i)17-s + (0.258 − 0.149i)19-s + (0.983 + 0.180i)21-s + (−1.44 + 0.832i)23-s + (−0.103 − 0.994i)27-s + 0.642i·29-s + (−0.998 − 0.576i)31-s + (0.129 − 0.243i)33-s + (0.414 + 0.718i)37-s + (−1.19 + 0.0412i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.998 - 0.0549i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.998 - 0.0549i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6345530712\)
\(L(\frac12)\) \(\approx\) \(0.6345530712\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0597 - 1.73i)T \)
5 \( 1 \)
7 \( 1 + (-0.567 + 2.58i)T \)
good11 \( 1 + (0.793 + 0.457i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.31iT - 13T^{2} \)
17 \( 1 + (-0.268 + 0.465i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.12 + 0.651i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.91 - 3.99i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.46iT - 29T^{2} \)
31 \( 1 + (5.56 + 3.21i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.52 - 4.36i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 - 8.22T + 43T^{2} \)
47 \( 1 + (-2.17 - 3.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.70 - 0.984i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.15 - 12.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.38 - 4.84i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.05 - 10.4i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.943iT - 71T^{2} \)
73 \( 1 + (8.85 + 5.11i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.71 + 9.90i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.35T + 83T^{2} \)
89 \( 1 + (-0.874 - 1.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.7iT - 97T^{2} \)
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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

−9.433403432681689275062917229756, −8.955524835916825128297420253142, −7.900977183514595293839147323215, −7.29715360428376586757795624101, −6.23320468405642290427832613235, −5.41891364959528820256189829131, −4.41061982768751309943905653106, −3.98036947427954122776446107741, −2.96093649580777173115534294772, −1.59602820781281079423993350700, 0.21005021570522893601256666549, 1.75843875672578954872063442092, 2.55713954803767385403075266859, 3.50753354513101344783164968780, 4.92162510989304197170608817320, 5.75731007220217196082878629366, 6.19567626056679730804895799359, 7.31573939121011223550620580979, 7.995184813092649063580564828635, 8.481086212007710301809424177330

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