Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.998 - 0.0618i$
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 − 3.41i·11-s + (−4 + 4i)13-s + 1.00·14-s − 1.00·16-s + (−3.41 + 3.41i)17-s − 2.82i·19-s + (2.41 + 2.41i)22-s + (0.828 + 0.828i)23-s − 5.65i·26-s + (−0.707 + 0.707i)28-s − 4.82·29-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 − 1.02i·11-s + (−1.10 + 1.10i)13-s + 0.267·14-s − 0.250·16-s + (−0.828 + 0.828i)17-s − 0.648i·19-s + (0.514 + 0.514i)22-s + (0.172 + 0.172i)23-s − 1.10i·26-s + (−0.133 + 0.133i)28-s − 0.896·29-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.998 - 0.0618i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.998 - 0.0618i)$
$L(1)$  $\approx$  $1.051247715$
$L(\frac12)$  $\approx$  $1.051247715$
$L(\frac{3}{2})$   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 \( 1 + 3.41iT - 11T^{2} \)
13 \( 1 + (4 - 4i)T - 13iT^{2} \)
17 \( 1 + (3.41 - 3.41i)T - 17iT^{2} \)
19 \( 1 + 2.82iT - 19T^{2} \)
23 \( 1 + (-0.828 - 0.828i)T + 23iT^{2} \)
29 \( 1 + 4.82T + 29T^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + (-2.24 - 2.24i)T + 37iT^{2} \)
41 \( 1 - 8.82iT - 41T^{2} \)
43 \( 1 + (-8.07 + 8.07i)T - 43iT^{2} \)
47 \( 1 + (0.757 - 0.757i)T - 47iT^{2} \)
53 \( 1 + (-5.41 - 5.41i)T + 53iT^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 - 9.89T + 61T^{2} \)
67 \( 1 + (5.58 + 5.58i)T + 67iT^{2} \)
71 \( 1 + 6.82iT - 71T^{2} \)
73 \( 1 + (-7.07 + 7.07i)T - 73iT^{2} \)
79 \( 1 + 5.65iT - 79T^{2} \)
83 \( 1 + (4.82 + 4.82i)T + 83iT^{2} \)
89 \( 1 - 8.82T + 89T^{2} \)
97 \( 1 + (-7.41 - 7.41i)T + 97iT^{2} \)
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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

−8.757701440729275528281114744054, −7.937080081456609917677242792860, −7.18531369384163984432314008640, −6.48965846482341436286266605069, −5.93128344784973761588199548937, −4.81338630475251227989444935888, −4.19048041013594969349771256930, −2.96963474987026152988058099722, −1.95536680568999105155571947792, −0.58102001948101867782320349448, 0.72753017720800673085444941335, 2.26292775782187156695250846905, 2.68227119872374468039129633600, 3.89513974211867058115027798370, 4.76303431715305975095088606269, 5.52811744282305580891745826786, 6.61898344630869156134064380585, 7.35762168038180591105661024173, 7.906110842146838160309692879030, 8.800819610362495586150622132263

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