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Properties

Label 2-280-40.29-c1-0-30
Degree $2$
Conductor $280$
Sign $0.997 - 0.0661i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
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  + (1.38 + 0.290i)2-s + 1.83·3-s + (1.83 + 0.805i)4-s + (−1.41 − 1.72i)5-s + (2.53 + 0.532i)6-s − i·7-s + (2.29 + 1.64i)8-s + 0.355·9-s + (−1.46 − 2.80i)10-s + 1.36i·11-s + (3.35 + 1.47i)12-s − 3.84·13-s + (0.290 − 1.38i)14-s + (−2.60 − 3.16i)15-s + (2.70 + 2.94i)16-s + 4.27i·17-s + ⋯
L(s)  = 1  + (0.978 + 0.205i)2-s + 1.05·3-s + (0.915 + 0.402i)4-s + (−0.634 − 0.772i)5-s + (1.03 + 0.217i)6-s − 0.377i·7-s + (0.812 + 0.582i)8-s + 0.118·9-s + (−0.462 − 0.886i)10-s + 0.411i·11-s + (0.968 + 0.425i)12-s − 1.06·13-s + (0.0777 − 0.369i)14-s + (−0.671 − 0.817i)15-s + (0.675 + 0.737i)16-s + 1.03i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.997 - 0.0661i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.997 - 0.0661i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.65594 + 0.0879735i\)
\(L(\frac12)\) \(\approx\) \(2.65594 + 0.0879735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.290i)T \)
5 \( 1 + (1.41 + 1.72i)T \)
7 \( 1 + iT \)
good3 \( 1 - 1.83T + 3T^{2} \)
11 \( 1 - 1.36iT - 11T^{2} \)
13 \( 1 + 3.84T + 13T^{2} \)
17 \( 1 - 4.27iT - 17T^{2} \)
19 \( 1 + 3.72iT - 19T^{2} \)
23 \( 1 - 5.92iT - 23T^{2} \)
29 \( 1 + 7.35iT - 29T^{2} \)
31 \( 1 + 2.11T + 31T^{2} \)
37 \( 1 + 2.33T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 0.234T + 43T^{2} \)
47 \( 1 + 9.73iT - 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 13.1iT - 59T^{2} \)
61 \( 1 - 6.91iT - 61T^{2} \)
67 \( 1 + 5.95T + 67T^{2} \)
71 \( 1 - 3.49T + 71T^{2} \)
73 \( 1 + 2.20iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + 6.90T + 83T^{2} \)
89 \( 1 - 8.59T + 89T^{2} \)
97 \( 1 + 9.01iT - 97T^{2} \)
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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

−12.07496039458556744925453716424, −11.27571622171673336095254446746, −9.894240692717249666153470780862, −8.781442522248126765275154543140, −7.80771624306004178610757914384, −7.22106910445236347109068356382, −5.60446985540022303600385921977, −4.44010124128021390542910957798, −3.58031374717520273186411770221, −2.18504979615010251217370342873, 2.48812871580836718015512098789, 3.12966082130200697577831901962, 4.35110494391409874313023056468, 5.71778585517271315392905296470, 7.02475920650110035633293229159, 7.76421754194436952157876634372, 8.949403234989166042030588989120, 10.13987838256847148066020230587, 11.08645613443753203799047239994, 12.00155687903897654053787030969

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