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Properties

Label 2-2240-5.4-c1-0-68
Degree $2$
Conductor $2240$
Sign $-0.956 - 0.290i$
Analytic cond. $17.8864$
Root an. cond. $4.22924$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.85i·3-s + (0.649 − 2.13i)5-s − i·7-s − 5.15·9-s + 4.61·11-s − 0.332i·13-s + (−6.11 − 1.85i)15-s − 7.60i·17-s − 1.28·19-s − 2.85·21-s + 3.71i·23-s + (−4.15 − 2.78i)25-s + 6.15i·27-s + 2.44·29-s − 8.55·31-s + ⋯
L(s)  = 1  − 1.64i·3-s + (0.290 − 0.956i)5-s − 0.377i·7-s − 1.71·9-s + 1.39·11-s − 0.0922i·13-s + (−1.57 − 0.479i)15-s − 1.84i·17-s − 0.294·19-s − 0.623·21-s + 0.773i·23-s + (−0.831 − 0.556i)25-s + 1.18i·27-s + 0.453·29-s − 1.53·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $-0.956 - 0.290i$
Analytic conductor: \(17.8864\)
Root analytic conductor: \(4.22924\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :1/2),\ -0.956 - 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.704091421\)
\(L(\frac12)\) \(\approx\) \(1.704091421\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.649 + 2.13i)T \)
7 \( 1 + iT \)
good3 \( 1 + 2.85iT - 3T^{2} \)
11 \( 1 - 4.61T + 11T^{2} \)
13 \( 1 + 0.332iT - 13T^{2} \)
17 \( 1 + 7.60iT - 17T^{2} \)
19 \( 1 + 1.28T + 19T^{2} \)
23 \( 1 - 3.71iT - 23T^{2} \)
29 \( 1 - 2.44T + 29T^{2} \)
31 \( 1 + 8.55T + 31T^{2} \)
37 \( 1 - 5.56iT - 37T^{2} \)
41 \( 1 - 7.71T + 41T^{2} \)
43 \( 1 + 9.71iT - 43T^{2} \)
47 \( 1 - 8.75iT - 47T^{2} \)
53 \( 1 + 12.2iT - 53T^{2} \)
59 \( 1 - 1.28T + 59T^{2} \)
61 \( 1 - 3.01T + 61T^{2} \)
67 \( 1 - 12.3iT - 67T^{2} \)
71 \( 1 + 2.99T + 71T^{2} \)
73 \( 1 + 12.2iT - 73T^{2} \)
79 \( 1 - 0.949T + 79T^{2} \)
83 \( 1 + 3.81iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 1.61iT - 97T^{2} \)
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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

−8.629833827024992388086304145662, −7.62417422588212256139764450626, −7.15855720651066476818807533759, −6.40975359044894432531062893295, −5.62081242297640004254376996001, −4.72388083385691711499795592998, −3.58151938081943079460056745533, −2.29575559977214116769719190059, −1.37638405053788023582895289753, −0.61165325579151750415129617715, 1.87878628048414817571971004181, 3.05623383318533117207517317979, 3.92971941239656351742430818631, 4.30642975715358414862048808037, 5.64130777817556716144881057325, 6.11282947956138069652736855013, 6.95168501264834315089674444179, 8.185181469944460080027209664861, 8.974560906178372124479724162832, 9.457446463323173776503561359635

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