Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 $
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

\( d \)  =  \(2\)
\( N \)  =  \(2100\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.900 - 0.435i$
motivic weight  =  \(1\)
character  :  $\chi_{2100} (1201, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2100,\ (\ :1/2),\ -0.900 - 0.435i)\)
\(L(1)\)  \(\approx\)  \(0.3859018741\)
\(L(\frac12)\)  \(\approx\)  \(0.3859018741\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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