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

Related objects

Downloads

Learn more about

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 + 0.585i·11-s + (−4 + 4i)13-s − 1.00·14-s − 1.00·16-s + (0.585 − 0.585i)17-s + 2.82i·19-s + (−0.414 − 0.414i)22-s + (4.82 + 4.82i)23-s − 5.65i·26-s + (0.707 − 0.707i)28-s − 0.828·29-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 + 0.176i·11-s + (−1.10 + 1.10i)13-s − 0.267·14-s − 0.250·16-s + (0.142 − 0.142i)17-s + 0.648i·19-s + (−0.0883 − 0.0883i)22-s + (1.00 + 1.00i)23-s − 1.10i·26-s + (0.133 − 0.133i)28-s − 0.153·29-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  =  $(2,\ 3150,\ (\ :1/2),\ -0.998 + 0.0618i)$
$L(1)$  $\approx$  $0.5667641680$
$L(\frac12)$  $\approx$  $0.5667641680$
$L(\frac{3}{2})$   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 \( 1 - 0.585iT - 11T^{2} \)
13 \( 1 + (4 - 4i)T - 13iT^{2} \)
17 \( 1 + (-0.585 + 0.585i)T - 17iT^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 + (-4.82 - 4.82i)T + 23iT^{2} \)
29 \( 1 + 0.828T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 + (6.24 + 6.24i)T + 37iT^{2} \)
41 \( 1 + 3.17iT - 41T^{2} \)
43 \( 1 + (6.07 - 6.07i)T - 43iT^{2} \)
47 \( 1 + (-9.24 + 9.24i)T - 47iT^{2} \)
53 \( 1 + (2.58 + 2.58i)T + 53iT^{2} \)
59 \( 1 + 2.82T + 59T^{2} \)
61 \( 1 + 9.89T + 61T^{2} \)
67 \( 1 + (8.41 + 8.41i)T + 67iT^{2} \)
71 \( 1 - 1.17iT - 71T^{2} \)
73 \( 1 + (7.07 - 7.07i)T - 73iT^{2} \)
79 \( 1 - 5.65iT - 79T^{2} \)
83 \( 1 + (0.828 + 0.828i)T + 83iT^{2} \)
89 \( 1 + 3.17T + 89T^{2} \)
97 \( 1 + (-4.58 - 4.58i)T + 97iT^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.115188106968736645452974274440, −8.335887891779964536809139244289, −7.37585080680054045077033100163, −7.11879010609637318839229481865, −6.11388273049151311751016322272, −5.27652924173928646324344956071, −4.66484310720473843057625193760, −3.58106677386434096493193452698, −2.33687392539710334173977200568, −1.46005722063724427455159793663, 0.21224958927974796384427538446, 1.36138840676851781010874336715, 2.66455459517936266052932394310, 3.17122841109996786359978097928, 4.47258105463508237614342821531, 5.01938473869581091185097159526, 6.07452319785739702541658928777, 7.08939389483204367684743247361, 7.58222819550727480380726735745, 8.425770819406378904364415093875

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