Properties

Degree $2$
Conductor $2100$
Sign $-0.900 + 0.435i$
Motivic weight $1$
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 − 5·13-s + (−2.12 − 3.67i)17-s + (−1.62 + 2.80i)19-s + (−2.62 − 0.358i)21-s + (3 − 5.19i)23-s − 0.999·27-s − 8.48·29-s + (2 + 3.46i)31-s + (−2.12 + 3.67i)33-s + (2.5 − 4.33i)37-s + (−2.5 − 4.33i)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 − 1.38·13-s + (−0.514 − 0.891i)17-s + (−0.371 + 0.644i)19-s + (−0.572 − 0.0782i)21-s + (0.625 − 1.08i)23-s − 0.192·27-s − 1.57·29-s + (0.359 + 0.622i)31-s + (−0.369 + 0.639i)33-s + (0.410 − 0.711i)37-s + (−0.400 − 0.693i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
Sign: $-0.900 + 0.435i$
Motivic weight: \(1\)
Character: $\chi_{2100} (1801, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2100,\ (\ :1/2),\ -0.900 + 0.435i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3859018741\)
\(L(\frac12)\) \(\approx\) \(0.3859018741\)
\(L(\frac{3}{2})\) 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 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (2.12 + 3.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.62 - 2.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.75T + 41T^{2} \)
43 \( 1 + 4.48T + 43T^{2} \)
47 \( 1 + (-0.878 + 1.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.12 + 3.67i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.878 + 1.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.86 + 11.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.48T + 71T^{2} \)
73 \( 1 + (-6.74 - 11.6i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.378 - 0.655i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 + (8.12 - 14.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 15.9T + 97T^{2} \)
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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

−9.519214687997076500780451483540, −9.033474557164664034037330048680, −8.081840514434395283440539221811, −7.09248772839407182755779570075, −6.59018359807203580378296515854, −5.37628810429703844266730570057, −4.76123984050685927758013431927, −3.85117540421915734118539905393, −2.72058292628786625949453293373, −2.03586962395108747571328974402, 0.12347801075692190203091457018, 1.46774573674487514944931020117, 2.74176656695706000903946859414, 3.59175019709731634243443303367, 4.44611033473386652078181611597, 5.63900081481304789680064521445, 6.43963969332001296391871019668, 7.12874541227980673146179098301, 7.76110197911715348682748249356, 8.699129912603561413400298632336

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