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Properties

Label 2-2100-21.5-c1-0-45
Degree $2$
Conductor $2100$
Sign $-0.444 + 0.895i$
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  + (1.5 − 0.866i)3-s + (−2 + 1.73i)7-s + (1.5 − 2.59i)9-s − 6.92i·13-s + (−3 − 1.73i)19-s + (−1.50 + 4.33i)21-s − 5.19i·27-s + (1.5 − 0.866i)31-s + (0.5 − 0.866i)37-s + (−5.99 − 10.3i)39-s − 13·43-s + (1.00 − 6.92i)49-s − 6·57-s + (7.5 + 4.33i)61-s + (1.5 + 7.79i)63-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.755 + 0.654i)7-s + (0.5 − 0.866i)9-s − 1.92i·13-s + (−0.688 − 0.397i)19-s + (−0.327 + 0.944i)21-s − 0.999i·27-s + (0.269 − 0.155i)31-s + (0.0821 − 0.142i)37-s + (−0.960 − 1.66i)39-s − 1.98·43-s + (0.142 − 0.989i)49-s − 0.794·57-s + (0.960 + 0.554i)61-s + (0.188 + 0.981i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.444 + 0.895i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.649613461\)
\(L(\frac12)\) \(\approx\) \(1.649613461\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (2 - 1.73i)T \)
good11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.92iT - 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3 + 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 13T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.5 - 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8 + 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-13.5 + 7.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 19.0iT - 97T^{2} \)
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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

−8.661724395501958977729491669741, −8.224772261083050511153160437764, −7.40052455641688474214672250805, −6.52068835648759771299220985608, −5.85448920140520726094720335725, −4.84418199301923632997843338166, −3.50203604629021938684752076343, −2.98811235638487877318818563393, −2.03649988431347988753783819853, −0.49510254617224935963253262031, 1.60237377675957337527355375719, 2.63318211974073032875383706191, 3.77652796553541269576553591449, 4.17659702734318322918859277210, 5.16528651561442689408882903949, 6.58243371987730664460637482408, 6.86072647951001461470110821494, 7.954247371253177825668735956615, 8.658800311390515134522224373433, 9.427166226359367537511948221315

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