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Properties

Label 2-3675-21.2-c0-0-2
Degree $2$
Conductor $3675$
Sign $0.126 - 0.991i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s + (0.499 − 0.866i)9-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)16-s + (−1.73 + i)17-s + 0.999i·27-s − 0.999·36-s + (1.73 + i)47-s − 0.999i·48-s + (0.999 − 1.73i)51-s + 0.999·64-s + (1.73 + 0.999i)68-s + (−1 + 1.73i)79-s + (−0.5 − 0.866i)81-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.5 − 0.866i)4-s + (0.499 − 0.866i)9-s + (0.866 + 0.499i)12-s + (−0.499 + 0.866i)16-s + (−1.73 + i)17-s + 0.999i·27-s − 0.999·36-s + (1.73 + i)47-s − 0.999i·48-s + (0.999 − 1.73i)51-s + 0.999·64-s + (1.73 + 0.999i)68-s + (−1 + 1.73i)79-s + (−0.5 − 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.126 - 0.991i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.126 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5057883487\)
\(L(\frac12)\) \(\approx\) \(0.5057883487\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 2iT - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.014305596728288564997868550168, −8.400282079088037896568499781843, −7.15992871073387845969347451222, −6.40488010956900813157396670089, −5.90451412257362294042994342512, −5.11463551762738628865296784253, −4.36988783741455222517272017635, −3.85837619821449973749196039083, −2.30439102216675380266862881058, −1.11557228854828374353590489623, 0.37646360222908505327163146353, 2.01247329896486790040110599672, 2.92387641823813717208499251324, 4.15967593893003488904474067750, 4.67969709517985621770026115620, 5.49483563171297673495130064646, 6.43902700047500532244408601960, 7.15416042612149016755022297401, 7.58511405441797821968322754305, 8.608612589234536758683088139332

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