Properties

Label 2-2100-5.4-c1-0-6
Degree $2$
Conductor $2100$
Sign $0.447 - 0.894i$
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  + i·3-s i·7-s − 9-s + 2·11-s − 4i·13-s + 6i·17-s − 6·19-s + 21-s + 8i·23-s i·27-s + 2·29-s + 10·31-s + 2i·33-s + 2i·37-s + 4·39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 0.603·11-s − 1.10i·13-s + 1.45i·17-s − 1.37·19-s + 0.218·21-s + 1.66i·23-s − 0.192i·27-s + 0.371·29-s + 1.79·31-s + 0.348i·33-s + 0.328i·37-s + 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.645847503\)
\(L(\frac12)\) \(\approx\) \(1.645847503\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 \)
7 \( 1 + iT \)
good11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 10T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.289569484418924082821522969970, −8.350391051818549967690285407541, −7.974112817924139248420238438662, −6.76981061066739201307051752352, −6.06903706652712303541500806622, −5.25868873853819181234085943899, −4.16879465314468951424228572460, −3.68250693458722001152303097672, −2.49989614625151099530791970929, −1.10373764308899807179398245954, 0.68420201427708263942146792344, 2.10877982190777344552363434709, 2.79355446296780662218616378875, 4.25382248683361944596631067508, 4.77609113716349184770249196678, 6.16528124488797946418264683660, 6.50593571263241933918818347566, 7.31094032507319518104193724301, 8.291409565170240599241614333002, 8.920605660482480825294125261409

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