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Properties

Label 1-3744-3744.565-r1-0-0
Degree $1$
Conductor $3744$
Sign $0.0167 - 0.999i$
Analytic cond. $402.348$
Root an. cond. $402.348$
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  + (−0.258 − 0.965i)5-s − 7-s + (−0.258 − 0.965i)11-s + (−0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s − i·23-s + (−0.866 + 0.5i)25-s + (0.258 + 0.965i)29-s + (0.866 + 0.5i)31-s + (0.258 + 0.965i)35-s + (−0.965 − 0.258i)37-s − 41-s + (−0.707 − 0.707i)43-s + (−0.866 + 0.5i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)5-s − 7-s + (−0.258 − 0.965i)11-s + (−0.5 + 0.866i)17-s + (−0.965 + 0.258i)19-s − i·23-s + (−0.866 + 0.5i)25-s + (0.258 + 0.965i)29-s + (0.866 + 0.5i)31-s + (0.258 + 0.965i)35-s + (−0.965 − 0.258i)37-s − 41-s + (−0.707 − 0.707i)43-s + (−0.866 + 0.5i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0167 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0167 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.0167 - 0.999i$
Analytic conductor: \(402.348\)
Root analytic conductor: \(402.348\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3744,\ (1:\ ),\ 0.0167 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3771410437 - 0.3835297393i\)
\(L(\frac12)\) \(\approx\) \(0.3771410437 - 0.3835297393i\)
\(L(1)\) \(\approx\) \(0.7096699272 - 0.09000267288i\)
\(L(1)\) \(\approx\) \(0.7096699272 - 0.09000267288i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
good5 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.258 - 0.965i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.965 + 0.258i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.258 + 0.965i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 + (-0.965 - 0.258i)T \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.258 - 0.965i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.62708283448397805825751668728, −18.03421821602129684103198441814, −17.32471927464419997480502686524, −16.50578169597359421837163618193, −15.7706836494166535768139151658, −15.140019492351708202572457317057, −14.763608384234419200496724995817, −13.58935721323312982292493384296, −13.32487753909434107144639960082, −12.281309531627535402373710708736, −11.80605879443832941976640241433, −10.90497000544944548245368756210, −10.10012208731963852041277185102, −9.87113567397247118172018212576, −8.812930059165715561380619130712, −8.03792644964594672305568426974, −7.08364266190581727921512665985, −6.68201611992415513898351657687, −6.11238631329451674179330877122, −4.87721824349990100645948361059, −4.25643696093788354674223664112, −3.32079913737292817031164013873, −2.59761455248439687951579187221, −2.03078639422423804256169447648, −0.42870158239421816597884076517, 0.16749894460620575686721425828, 1.18370246048176122102536716606, 2.05251927330664544944310627505, 3.297274513060446556338911354742, 3.667906192133490767939613367092, 4.67919492292869785764374538476, 5.43571764246859739043641912011, 6.19451353785756778728662438465, 6.82160847336562780697793842499, 7.87587572451701764004218114482, 8.63172237026141780026000789518, 8.922622334663137470819763138388, 9.9468735857507478551138962339, 10.56013159745095968609176704757, 11.369643590097538059760469059159, 12.27913082969628997324409849649, 12.68618173083556848730648335176, 13.45045628734575473297025081016, 13.8727057975981241716841717642, 15.092526757513820314280535108826, 15.64635061919061585686671398784, 16.20065154320404259532663130327, 16.91115739318090493704697144286, 17.32576847700504586119903974898, 18.36774111921215790026601584971

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