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Properties

Label 2-2100-35.12-c1-0-0
Degree $2$
Conductor $2100$
Sign $-0.214 - 0.976i$
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 + (−2.42 − 1.05i)7-s + (−0.866 − 0.499i)9-s + (−1.30 − 2.26i)11-s + (−2.50 − 2.50i)13-s + (1.07 + 0.287i)17-s + (−3.66 + 6.35i)19-s + (−1.65 + 2.06i)21-s + (0.398 + 1.48i)23-s + (−0.707 + 0.707i)27-s + 3.81i·29-s + (6.66 − 3.84i)31-s + (−2.52 + 0.676i)33-s + (−3.25 + 0.871i)37-s + (−3.06 + 1.76i)39-s + ⋯
L(s)  = 1  + (0.149 − 0.557i)3-s + (−0.916 − 0.400i)7-s + (−0.288 − 0.166i)9-s + (−0.393 − 0.682i)11-s + (−0.693 − 0.693i)13-s + (0.259 + 0.0696i)17-s + (−0.841 + 1.45i)19-s + (−0.360 + 0.451i)21-s + (0.0831 + 0.310i)23-s + (−0.136 + 0.136i)27-s + 0.708i·29-s + (1.19 − 0.690i)31-s + (−0.439 + 0.117i)33-s + (−0.534 + 0.143i)37-s + (−0.490 + 0.283i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.214 - 0.976i$
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: \((2,\ 2100,\ (\ :1/2),\ -0.214 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3650128062\)
\(L(\frac12)\) \(\approx\) \(0.3650128062\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.42 + 1.05i)T \)
good11 \( 1 + (1.30 + 2.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.50 + 2.50i)T + 13iT^{2} \)
17 \( 1 + (-1.07 - 0.287i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.66 - 6.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.398 - 1.48i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 3.81iT - 29T^{2} \)
31 \( 1 + (-6.66 + 3.84i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.25 - 0.871i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 8.65iT - 41T^{2} \)
43 \( 1 + (0.817 - 0.817i)T - 43iT^{2} \)
47 \( 1 + (-2.71 - 10.1i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.0300 - 0.00805i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.72 + 11.6i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.32 - 4.80i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.42 - 9.05i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 3.73T + 71T^{2} \)
73 \( 1 + (2.12 - 7.93i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.44 + 3.72i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.33 + 9.33i)T + 83iT^{2} \)
89 \( 1 + (-5.56 + 9.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.97 - 1.97i)T - 97iT^{2} \)
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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

−9.378253574193495075562390761033, −8.293418508450763695129495786858, −7.890934193621495141924720555046, −6.99609511955482515953356151165, −6.17224768853613200992334248185, −5.61284599201298167728457868495, −4.38166374998888361801656411920, −3.32809217069104789664680076245, −2.67688149366413468283423934172, −1.23879703960062198682132319006, 0.12782694352858452050349779325, 2.21742180417979864767446462247, 2.85763927816090035112353609363, 4.04750578128925542473383670190, 4.77594828991185276300110799261, 5.59499884494858131235172481399, 6.71857878130206094977123992841, 7.09822302178736753989824357939, 8.321410753609811326154890135381, 9.008007463988525716031455877241

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