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Properties

Label 2-60e2-3.2-c0-0-3
Degree $2$
Conductor $3600$
Sign $0.577 + 0.816i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
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  + 7-s − 1.41i·11-s + 13-s − 1.41i·17-s − 19-s − 1.41i·23-s + 1.41i·29-s − 31-s − 43-s − 1.41i·47-s + 1.41i·59-s + 61-s + 67-s − 1.41i·77-s + 1.41i·83-s + ⋯
L(s)  = 1  + 7-s − 1.41i·11-s + 13-s − 1.41i·17-s − 19-s − 1.41i·23-s + 1.41i·29-s − 31-s − 43-s − 1.41i·47-s + 1.41i·59-s + 61-s + 67-s − 1.41i·77-s + 1.41i·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.577 + 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.400968745\)
\(L(\frac12)\) \(\approx\) \(1.400968745\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - T + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−8.638500256103326820405106902297, −8.112920924487967028762584241507, −7.06957455539179760687525647156, −6.44428374035513674183502844617, −5.50673453653664649039742821198, −4.93607296628208658586609165469, −3.94764311166269882958010416617, −3.12468352684651830307925912594, −2.06006397745463030029900780347, −0.854814433917733532252852419903, 1.61245723757089528019772218303, 2.02664624170740135540600264529, 3.58369756359856413017064034223, 4.22718171592799916570217338214, 4.98561557376672697548185602270, 5.90005240869906844789575128064, 6.56126826123472603347650751197, 7.55525502599468323412494881251, 8.055589366567735307324079009058, 8.710564308136195994525248405738

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