Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.405 - 0.914i$
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 + (0.295 − 2.62i)7-s − 0.999i·8-s + (0.570 + 0.329i)11-s + 6.13i·13-s + (−1.05 − 2.42i)14-s + (−0.5 − 0.866i)16-s + (−2.43 + 4.22i)17-s + (−6.30 + 3.63i)19-s + 0.659·22-s + (−3.98 + 2.29i)23-s + (3.06 + 5.31i)26-s + (−2.12 − 1.57i)28-s + 8.09i·29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.111 − 0.993i)7-s − 0.353i·8-s + (0.172 + 0.0993i)11-s + 1.70i·13-s + (−0.282 − 0.648i)14-s + (−0.125 − 0.216i)16-s + (−0.591 + 1.02i)17-s + (−1.44 + 0.834i)19-s + 0.140·22-s + (−0.830 + 0.479i)23-s + (0.601 + 1.04i)26-s + (−0.402 − 0.296i)28-s + 1.50i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.405 - 0.914i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.405 - 0.914i)$
$L(1)$  $\approx$  $1.695110674$
$L(\frac12)$  $\approx$  $1.695110674$
$L(\frac{3}{2})$   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 \( 1 + (-0.295 + 2.62i)T \)
good11 \( 1 + (-0.570 - 0.329i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.13iT - 13T^{2} \)
17 \( 1 + (2.43 - 4.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.30 - 3.63i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.98 - 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 8.09iT - 29T^{2} \)
31 \( 1 + (-0.759 - 0.438i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.05 - 8.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.25T + 41T^{2} \)
43 \( 1 - 9.03T + 43T^{2} \)
47 \( 1 + (6.00 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.5 - 6.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.06 + 7.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.0618 - 0.0357i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.666 + 1.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.60iT - 71T^{2} \)
73 \( 1 + (-2.44 - 1.41i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.88 + 5.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.44T + 83T^{2} \)
89 \( 1 + (2.66 + 4.61i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 11.4iT - 97T^{2} \)
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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

−8.756372791010738458149131172985, −8.161310406112259171753584258452, −6.97137461650393653945385345234, −6.65234809333327367885385850025, −5.82365678645640629144571543193, −4.62242699260344212272286298189, −4.15225027248829121818664562539, −3.54688575291035924297328131377, −2.07459346000325634827607548113, −1.47895866596135722576549134044, 0.39455396283672155178036625955, 2.38449917694813693389723312879, 2.68143952384469683025584415671, 3.99588806830018783637086016257, 4.71533639232852958687022711660, 5.63200746154573676982488776618, 6.06308523506698412161116927558, 6.92507113285996012079857179232, 7.87170464116322797606848559714, 8.383814203472661722606460825291

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