Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $-0.929 + 0.368i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.24 + 0.671i)2-s − 3-s + (1.09 − 1.67i)4-s + (1.24 − 0.671i)6-s − 4.68i·7-s + (−0.244 + 2.81i)8-s + 9-s − 2.29i·11-s + (−1.09 + 1.67i)12-s − 4.97·13-s + (3.14 + 5.83i)14-s + (−1.58 − 3.67i)16-s + 2.97i·17-s + (−1.24 + 0.671i)18-s + 2.68i·19-s + ⋯
L(s)  = 1  + (−0.880 + 0.474i)2-s − 0.577·3-s + (0.549 − 0.835i)4-s + (0.508 − 0.274i)6-s − 1.77i·7-s + (−0.0864 + 0.996i)8-s + 0.333·9-s − 0.691i·11-s + (−0.317 + 0.482i)12-s − 1.38·13-s + (0.840 + 1.55i)14-s + (−0.396 − 0.917i)16-s + 0.722i·17-s + (−0.293 + 0.158i)18-s + 0.616i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.929 + 0.368i$
motivic weight  =  \(1\)
character  :  $\chi_{600} (349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :1/2),\ -0.929 + 0.368i)\)
\(L(1)\)  \(\approx\)  \(0.0386860 - 0.202769i\)
\(L(\frac12)\)  \(\approx\)  \(0.0386860 - 0.202769i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (1.24 - 0.671i)T \)
3 \( 1 + T \)
5 \( 1 \)
good7 \( 1 + 4.68iT - 7T^{2} \)
11 \( 1 + 2.29iT - 11T^{2} \)
13 \( 1 + 4.97T + 13T^{2} \)
17 \( 1 - 2.97iT - 17T^{2} \)
19 \( 1 - 2.68iT - 19T^{2} \)
23 \( 1 - 2.68iT - 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 6.97T + 31T^{2} \)
37 \( 1 + 4.39T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 - 9.37T + 43T^{2} \)
47 \( 1 + 7.27iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 1.70iT - 59T^{2} \)
61 \( 1 + 4.58iT - 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 0.585T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 1.02T + 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 + 3.37T + 89T^{2} \)
97 \( 1 - 3.95iT - 97T^{2} \)
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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

−10.33167703261758230817142898171, −9.600100498880543140923733283646, −8.393972131618795726704724514302, −7.40233120213116686245365138930, −7.02075718292190778703267685126, −5.88253842929194725291443068078, −4.89450136833995656433304553409, −3.61138327812209258684653296657, −1.59766349238880785080994304301, −0.15903604181634495935897673885, 2.02945077535826425175472285014, 2.84517853161744640264327861742, 4.63920355519819174369012170356, 5.57605520652319942218035698053, 6.77877885429962459328927523297, 7.53337282257966056634324225003, 8.706128994632291139668886243773, 9.380329199787679926697331365580, 10.02026040176976769975526345745, 11.11157654784755477245954175295

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