Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.153 + 0.988i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (0.397 − 2.61i)7-s + 0.999i·8-s + (0.429 − 0.248i)11-s − 2.74i·13-s + (1.65 − 2.06i)14-s + (−0.5 + 0.866i)16-s + (−1.82 − 3.16i)17-s + (3.12 + 1.80i)19-s + 0.496·22-s + (−5.56 − 3.21i)23-s + (1.37 − 2.37i)26-s + (2.46 − 0.963i)28-s − 8.87i·29-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.150 − 0.988i)7-s + 0.353i·8-s + (0.129 − 0.0748i)11-s − 0.761i·13-s + (0.441 − 0.552i)14-s + (−0.125 + 0.216i)16-s + (−0.443 − 0.768i)17-s + (0.716 + 0.413i)19-s + 0.105·22-s + (−1.16 − 0.669i)23-s + (0.269 − 0.466i)26-s + (0.465 − 0.182i)28-s − 1.64i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.153 + 0.988i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.153 + 0.988i)$
$L(1)$  $\approx$  $1.942000641$
$L(\frac12)$  $\approx$  $1.942000641$
$L(\frac{3}{2})$   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 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.397 + 2.61i)T \)
good11 \( 1 + (-0.429 + 0.248i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.74iT - 13T^{2} \)
17 \( 1 + (1.82 + 3.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.12 - 1.80i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.56 + 3.21i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.87iT - 29T^{2} \)
31 \( 1 + (6.90 - 3.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.14 - 1.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.22T + 41T^{2} \)
43 \( 1 - 2.22T + 43T^{2} \)
47 \( 1 + (-3.27 + 5.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.72 - 3.88i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.05 + 5.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.24 - 1.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.08 + 7.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + (-11.3 + 6.53i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.44 + 7.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.79T + 83T^{2} \)
89 \( 1 + (-0.743 + 1.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.05iT - 97T^{2} \)
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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

−8.190685541672051925820389829416, −7.71969785384552561855560812811, −7.00589382893884933155204405368, −6.23346596539712566852421879342, −5.43769949775601954243732874043, −4.62115630234404834112431477718, −3.87659021623019072432109275424, −3.09640864622291306231111515005, −1.91926091599887961181684838816, −0.45490371336672031036673408736, 1.55703301226617277295615208432, 2.25226242577471558781924458453, 3.34508689627595862108895436981, 4.11013940402753398399992093187, 5.05745473592747699041287815049, 5.69435451325416668313616470251, 6.42068924201038995465996506432, 7.25143961208246303719748664039, 8.123873333046920006671515064872, 9.111686413948204252088858031275

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