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Properties

Label 1-3648-3648.1181-r0-0-0
Degree $1$
Conductor $3648$
Sign $-0.982 + 0.183i$
Analytic cond. $16.9412$
Root an. cond. $16.9412$
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  + (−0.0436 − 0.999i)5-s + (0.258 + 0.965i)7-s + (−0.793 − 0.608i)11-s + (0.887 − 0.461i)13-s + (0.984 − 0.173i)17-s + (−0.906 − 0.422i)23-s + (−0.996 + 0.0871i)25-s + (−0.537 + 0.843i)29-s + (−0.5 + 0.866i)31-s + (0.953 − 0.300i)35-s + (−0.382 − 0.923i)37-s + (−0.996 − 0.0871i)41-s + (−0.999 + 0.0436i)43-s + (−0.984 − 0.173i)47-s + (−0.866 + 0.5i)49-s + ⋯
L(s)  = 1  + (−0.0436 − 0.999i)5-s + (0.258 + 0.965i)7-s + (−0.793 − 0.608i)11-s + (0.887 − 0.461i)13-s + (0.984 − 0.173i)17-s + (−0.906 − 0.422i)23-s + (−0.996 + 0.0871i)25-s + (−0.537 + 0.843i)29-s + (−0.5 + 0.866i)31-s + (0.953 − 0.300i)35-s + (−0.382 − 0.923i)37-s + (−0.996 − 0.0871i)41-s + (−0.999 + 0.0436i)43-s + (−0.984 − 0.173i)47-s + (−0.866 + 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $-0.982 + 0.183i$
Analytic conductor: \(16.9412\)
Root analytic conductor: \(16.9412\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (1181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3648,\ (0:\ ),\ -0.982 + 0.183i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02158158854 - 0.2329164234i\)
\(L(\frac12)\) \(\approx\) \(0.02158158854 - 0.2329164234i\)
\(L(1)\) \(\approx\) \(0.8620260579 - 0.1420425848i\)
\(L(1)\) \(\approx\) \(0.8620260579 - 0.1420425848i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (-0.0436 - 0.999i)T \)
7 \( 1 + (0.258 + 0.965i)T \)
11 \( 1 + (-0.793 - 0.608i)T \)
13 \( 1 + (0.887 - 0.461i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
23 \( 1 + (-0.906 - 0.422i)T \)
29 \( 1 + (-0.537 + 0.843i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.382 - 0.923i)T \)
41 \( 1 + (-0.996 - 0.0871i)T \)
43 \( 1 + (-0.999 + 0.0436i)T \)
47 \( 1 + (-0.984 - 0.173i)T \)
53 \( 1 + (0.737 - 0.675i)T \)
59 \( 1 + (0.843 - 0.537i)T \)
61 \( 1 + (-0.999 - 0.0436i)T \)
67 \( 1 + (-0.537 + 0.843i)T \)
71 \( 1 + (-0.906 + 0.422i)T \)
73 \( 1 + (0.996 + 0.0871i)T \)
79 \( 1 + (0.642 + 0.766i)T \)
83 \( 1 + (-0.608 - 0.793i)T \)
89 \( 1 + (-0.996 + 0.0871i)T \)
97 \( 1 + (0.173 + 0.984i)T \)
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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

−18.89580065600432831419975779253, −18.29820387038459289796373451861, −17.865196702897435582981635817037, −16.87536600237214926974958926741, −16.450660593389204906880203659464, −15.33331133990450562773339193764, −15.07783215082980507265782126779, −14.08651255611557923578374315069, −13.64255788285216769682894194189, −13.031505017036041740031494999157, −11.84999274866403743342415609117, −11.4445287835963897202820576559, −10.498248902783235016897255569, −10.18119539175863410683014996205, −9.46811329671057824441698011021, −8.1616505552533959206939084127, −7.76887523175187558293761558252, −7.06518723913644917299255915178, −6.314476328829119399826667668317, −5.57061016218210588377017616144, −4.55486769106991907043899126115, −3.76677986190661588863720427883, −3.22914715162508122427622302462, −2.10040577960203377566357106485, −1.42354447141237268762755978345, 0.06457203100052346174468028457, 1.28048339849927070829263319371, 1.98996459284605806190874170631, 3.10978890489882358647464435693, 3.71486101761171009296279090609, 4.89458219013005977454868657363, 5.48773108777884351268385852924, 5.818799908568662993196090415503, 6.979648050086371522429718963533, 8.14041625580550956176159405708, 8.31722979220606136645240529885, 9.009621098875023609070342147617, 9.890438037939448194307948376502, 10.63930720966968079177916688292, 11.479116672647234373493050352883, 12.14093529980635113568907800748, 12.75557050759944176121798671515, 13.338124680274631088441249616337, 14.162891882060327163843468989506, 14.9260370311960877961919412240, 15.77224575745324638911979065872, 16.2080235396850048141424287034, 16.69936408204903309865160061752, 17.86351004569223006306156492696, 18.22292246143245500973415164353

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