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Properties

Label 2-2100-35.12-c1-0-8
Degree $2$
Conductor $2100$
Sign $0.939 - 0.342i$
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 + (−2.54 − 0.728i)7-s + (−0.866 − 0.499i)9-s + (2.14 + 3.71i)11-s + (−3.22 − 3.22i)13-s + (3.44 + 0.923i)17-s + (3.13 − 5.42i)19-s + (1.36 − 2.26i)21-s + (−0.749 − 2.79i)23-s + (0.707 − 0.707i)27-s + 2.84i·29-s + (−1.34 + 0.775i)31-s + (−4.14 + 1.11i)33-s + (5.50 − 1.47i)37-s + (3.95 − 2.28i)39-s + ⋯
L(s)  = 1  + (−0.149 + 0.557i)3-s + (−0.961 − 0.275i)7-s + (−0.288 − 0.166i)9-s + (0.646 + 1.12i)11-s + (−0.895 − 0.895i)13-s + (0.835 + 0.223i)17-s + (0.719 − 1.24i)19-s + (0.297 − 0.494i)21-s + (−0.156 − 0.583i)23-s + (0.136 − 0.136i)27-s + 0.527i·29-s + (−0.241 + 0.139i)31-s + (−0.721 + 0.193i)33-s + (0.905 − 0.242i)37-s + (0.633 − 0.365i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.403723839\)
\(L(\frac12)\) \(\approx\) \(1.403723839\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.54 + 0.728i)T \)
good11 \( 1 + (-2.14 - 3.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.22 + 3.22i)T + 13iT^{2} \)
17 \( 1 + (-3.44 - 0.923i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.13 + 5.42i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.749 + 2.79i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 2.84iT - 29T^{2} \)
31 \( 1 + (1.34 - 0.775i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.50 + 1.47i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 10.7iT - 41T^{2} \)
43 \( 1 + (-2.38 + 2.38i)T - 43iT^{2} \)
47 \( 1 + (2.22 + 8.28i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-8.76 - 2.34i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.820 - 1.42i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.94 - 5.16i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.19 - 4.46i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + (2.40 - 8.97i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.68 + 3.86i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.32 - 3.32i)T + 83iT^{2} \)
89 \( 1 + (-8.91 + 15.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.100 + 0.100i)T - 97iT^{2} \)
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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

−9.406417479910108497050253122793, −8.497502604619422101651838116220, −7.34655373434721766427795233805, −6.97318835449571939776343537230, −5.91989577210412557764396996152, −5.08346183220785424653715163215, −4.29594060606145824196099735982, −3.33493517237171502717041925041, −2.50061790519154509930784407611, −0.76961666812455136519982170449, 0.796723155534329631027161567415, 2.11475313684536647366260825713, 3.23484392959023240447152626958, 3.94279065382957259972195492510, 5.35499226475760554705592687685, 5.95609864167031204972048639890, 6.65232049320612520215211101701, 7.49152337075648742507987972440, 8.169840865991385985460088831717, 9.276183735202287068370950548529

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