Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.0618 - 0.998i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + 0.828i·11-s + (−0.585 + 0.585i)13-s + 1.00·14-s − 1.00·16-s + (4.41 − 4.41i)17-s + 7.07i·19-s + (−0.585 − 0.585i)22-s + (−0.828 − 0.828i)23-s − 0.828i·26-s + (−0.707 + 0.707i)28-s − 8.82·29-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + 0.249i·11-s + (−0.162 + 0.162i)13-s + 0.267·14-s − 0.250·16-s + (1.07 − 1.07i)17-s + 1.62i·19-s + (−0.124 − 0.124i)22-s + (−0.172 − 0.172i)23-s − 0.162i·26-s + (−0.133 + 0.133i)28-s − 1.63·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0618 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.0618 - 0.998i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.0618 - 0.998i)$
$L(1)$  $\approx$  $1.061958238$
$L(\frac12)$  $\approx$  $1.061958238$
$L(\frac{3}{2})$   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 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 - 0.828iT - 11T^{2} \)
13 \( 1 + (0.585 - 0.585i)T - 13iT^{2} \)
17 \( 1 + (-4.41 + 4.41i)T - 17iT^{2} \)
19 \( 1 - 7.07iT - 19T^{2} \)
23 \( 1 + (0.828 + 0.828i)T + 23iT^{2} \)
29 \( 1 + 8.82T + 29T^{2} \)
31 \( 1 + 3.17T + 31T^{2} \)
37 \( 1 + (-8.07 - 8.07i)T + 37iT^{2} \)
41 \( 1 + 0.828iT - 41T^{2} \)
43 \( 1 + (-3.58 + 3.58i)T - 43iT^{2} \)
47 \( 1 + (-3.58 + 3.58i)T - 47iT^{2} \)
53 \( 1 + (-0.828 - 0.828i)T + 53iT^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 9.41T + 61T^{2} \)
67 \( 1 + (2.41 + 2.41i)T + 67iT^{2} \)
71 \( 1 - 3.75iT - 71T^{2} \)
73 \( 1 + (6 - 6i)T - 73iT^{2} \)
79 \( 1 - 9.31iT - 79T^{2} \)
83 \( 1 + (-1.65 - 1.65i)T + 83iT^{2} \)
89 \( 1 - 9.31T + 89T^{2} \)
97 \( 1 + (-0.242 - 0.242i)T + 97iT^{2} \)
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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

−8.903981985556281871466977344809, −7.88637565434205669242246254598, −7.55732240145586446133302422576, −6.75845001373547099228332786623, −5.84604812713721242148296533872, −5.31623076142919796900468160979, −4.21574381892640133108339444670, −3.37351232724570179105632955338, −2.15088657271021988618501190948, −1.00004516500147651667628390211, 0.46611510454590898647007515716, 1.75725168677132275018266508608, 2.74990076614155504863510322201, 3.56550488057576299344553345313, 4.43640181956095379047313711904, 5.56815889093120248179875097669, 6.13306589818614387452077428757, 7.30873197142435606293218073849, 7.70413223610336726068481374536, 8.669827660139424891742561325244

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