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Properties

Label 2-2100-7.2-c1-0-6
Degree $2$
Conductor $2100$
Sign $-0.198 - 0.980i$
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  + (0.5 + 0.866i)3-s + (−1.62 − 2.09i)7-s + (−0.499 + 0.866i)9-s + (2.12 + 3.67i)11-s − 3.24·13-s + (2.12 + 3.67i)17-s + (3.5 − 6.06i)19-s + (0.999 − 2.44i)21-s + (−2.12 + 3.67i)23-s − 0.999·27-s − 1.75·29-s + (4.74 + 8.21i)31-s + (−2.12 + 3.67i)33-s + (1.62 − 2.80i)37-s + (−1.62 − 2.80i)39-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.612 − 0.790i)7-s + (−0.166 + 0.288i)9-s + (0.639 + 1.10i)11-s − 0.899·13-s + (0.514 + 0.891i)17-s + (0.802 − 1.39i)19-s + (0.218 − 0.534i)21-s + (−0.442 + 0.766i)23-s − 0.192·27-s − 0.326·29-s + (0.851 + 1.47i)31-s + (−0.369 + 0.639i)33-s + (0.266 − 0.461i)37-s + (−0.259 − 0.449i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(16.7685\)
Root analytic conductor: \(4.09494\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2100} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.422361592\)
\(L(\frac12)\) \(\approx\) \(1.422361592\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (1.62 + 2.09i)T \)
good11 \( 1 + (-2.12 - 3.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.24T + 13T^{2} \)
17 \( 1 + (-2.12 - 3.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.12 - 3.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.75T + 29T^{2} \)
31 \( 1 + (-4.74 - 8.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.62 + 2.80i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 3.24T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.24 - 7.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.12 - 8.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.24 - 3.88i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.62 + 4.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (-4.62 - 8.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (-5.12 + 8.87i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.485T + 97T^{2} \)
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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

−9.449937924429776091240913661531, −8.735708372373216125195299240714, −7.52699384179861080614196547279, −7.19760228380596709297744383965, −6.27583961983357667543460389962, −5.12316882407980880135415851459, −4.42227669670630140205377249738, −3.59446645882664529056698809555, −2.68513851625811618048383748327, −1.32589602619959694192497166478, 0.50059021436165801319136165257, 1.97301262746325147740252826997, 2.98899934642563813287530544193, 3.65637864326432052291630318648, 5.01041680528625695505036018906, 5.89725436505570331867406412166, 6.42056997436564638370106393896, 7.39824965197648673155503723566, 8.164821109791289168986225068145, 8.780592769055256132146450953790

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