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Properties

Label 2-2100-5.4-c1-0-19
Degree $2$
Conductor $2100$
Sign $-0.894 - 0.447i$
Analytic cond. $16.7685$
Root an. cond. $4.09494$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s + i·7-s − 9-s − 11-s − 2i·13-s − 6·19-s + 21-s + i·23-s + i·27-s − 29-s − 2·31-s + i·33-s + 7i·37-s − 2·39-s − 8·41-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 0.301·11-s − 0.554i·13-s − 1.37·19-s + 0.218·21-s + 0.208i·23-s + 0.192i·27-s − 0.185·29-s − 0.359·31-s + 0.174i·33-s + 1.15i·37-s − 0.320·39-s − 1.24·41-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.894 - 0.447i$
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: \(1\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + 14iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 3iT - 67T^{2} \)
71 \( 1 + 9T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + T + 79T^{2} \)
83 \( 1 - 2iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.437607445892502240754332991967, −8.054314537523750635667214948919, −7.03677758921550525489848994865, −6.36131518541810305036280857163, −5.54235321092172402621071320641, −4.72152396439568378696730208417, −3.53913857488412270655402271833, −2.57639747745573851900917073996, −1.59015895549160105601758196548, 0, 1.77147715172929459234084824411, 2.90236480782709475829090676440, 4.00026074984720063504882424907, 4.54705737883419390992021308708, 5.55620179496014680898794398032, 6.38985305186693732324144016934, 7.21360091994638388362535412982, 8.083623376991630112615036332161, 8.888228717636609748923984109824

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