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Properties

Label 2-2100-35.3-c1-0-0
Degree $2$
Conductor $2100$
Sign $-0.992 + 0.119i$
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  + (0.258 + 0.965i)3-s + (−1.80 − 1.93i)7-s + (−0.866 + 0.499i)9-s + (2.59 − 4.50i)11-s + (−4.04 + 4.04i)13-s + (−1.22 + 0.327i)17-s + (3.15 + 5.45i)19-s + (1.40 − 2.24i)21-s + (−0.650 + 2.42i)23-s + (−0.707 − 0.707i)27-s − 6.75i·29-s + (−8.26 − 4.77i)31-s + (5.02 + 1.34i)33-s + (−5.80 − 1.55i)37-s + (−4.95 − 2.86i)39-s + ⋯
L(s)  = 1  + (0.149 + 0.557i)3-s + (−0.680 − 0.732i)7-s + (−0.288 + 0.166i)9-s + (0.783 − 1.35i)11-s + (−1.12 + 1.12i)13-s + (−0.296 + 0.0795i)17-s + (0.722 + 1.25i)19-s + (0.307 − 0.488i)21-s + (−0.135 + 0.506i)23-s + (−0.136 − 0.136i)27-s − 1.25i·29-s + (−1.48 − 0.857i)31-s + (0.874 + 0.234i)33-s + (−0.953 − 0.255i)37-s + (−0.793 − 0.458i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.992 + 0.119i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.992 + 0.119i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1042626559\)
\(L(\frac12)\) \(\approx\) \(0.1042626559\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 \)
7 \( 1 + (1.80 + 1.93i)T \)
good11 \( 1 + (-2.59 + 4.50i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.04 - 4.04i)T - 13iT^{2} \)
17 \( 1 + (1.22 - 0.327i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.15 - 5.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.650 - 2.42i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 6.75iT - 29T^{2} \)
31 \( 1 + (8.26 + 4.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.80 + 1.55i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.93iT - 41T^{2} \)
43 \( 1 + (2.26 + 2.26i)T + 43iT^{2} \)
47 \( 1 + (0.418 - 1.56i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (9.35 - 2.50i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.58 - 9.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.32 + 0.762i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.699 + 2.60i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 11.6T + 71T^{2} \)
73 \( 1 + (-2.03 - 7.59i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.49 + 0.861i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.17 + 5.17i)T - 83iT^{2} \)
89 \( 1 + (3.44 + 5.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.2 + 12.2i)T + 97iT^{2} \)
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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

−9.547847942955196456368235844322, −8.959344998605526070062353921519, −7.922378017042472410586228611485, −7.22732639474523676162662804883, −6.27688805447838103661125406636, −5.66373672734442831200840387731, −4.41751044985639565455976845426, −3.80595090590000683716704451111, −3.03707286911065432925091953769, −1.61032344415539648869747969261, 0.03394857341259606164037171005, 1.72339748998001556285578521534, 2.67572743233251748247592872121, 3.48502399137193739564048745120, 4.91617809591495327306884107791, 5.36142861091058898257381330869, 6.70248335786608015199753056394, 6.97611819616991224574552524304, 7.78456165498280437086417216718, 8.929942022604712715584877578270

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