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Properties

Label 2-287-41.32-c1-0-11
Degree $2$
Conductor $287$
Sign $0.986 - 0.166i$
Analytic cond. $2.29170$
Root an. cond. $1.51383$
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  + 0.447i·2-s + (1.09 − 1.09i)3-s + 1.79·4-s + 1.12i·5-s + (0.488 + 0.488i)6-s + (0.707 − 0.707i)7-s + 1.70i·8-s + 0.622i·9-s − 0.504·10-s + (−1.42 + 1.42i)11-s + (1.96 − 1.96i)12-s + (1.34 − 1.34i)13-s + (0.316 + 0.316i)14-s + (1.22 + 1.22i)15-s + 2.83·16-s + (−2.21 − 2.21i)17-s + ⋯
L(s)  = 1  + 0.316i·2-s + (0.629 − 0.629i)3-s + 0.899·4-s + 0.503i·5-s + (0.199 + 0.199i)6-s + (0.267 − 0.267i)7-s + 0.601i·8-s + 0.207i·9-s − 0.159·10-s + (−0.428 + 0.428i)11-s + (0.566 − 0.566i)12-s + (0.373 − 0.373i)13-s + (0.0846 + 0.0846i)14-s + (0.317 + 0.317i)15-s + 0.709·16-s + (−0.536 − 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.986 - 0.166i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.986 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84855 + 0.154864i\)
\(L(\frac12)\) \(\approx\) \(1.84855 + 0.154864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (6.15 + 1.77i)T \)
good2 \( 1 - 0.447iT - 2T^{2} \)
3 \( 1 + (-1.09 + 1.09i)T - 3iT^{2} \)
5 \( 1 - 1.12iT - 5T^{2} \)
11 \( 1 + (1.42 - 1.42i)T - 11iT^{2} \)
13 \( 1 + (-1.34 + 1.34i)T - 13iT^{2} \)
17 \( 1 + (2.21 + 2.21i)T + 17iT^{2} \)
19 \( 1 + (3.80 + 3.80i)T + 19iT^{2} \)
23 \( 1 + 8.28T + 23T^{2} \)
29 \( 1 + (-6.32 + 6.32i)T - 29iT^{2} \)
31 \( 1 - 2.36T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
43 \( 1 - 6.75iT - 43T^{2} \)
47 \( 1 + (-5.77 - 5.77i)T + 47iT^{2} \)
53 \( 1 + (1.80 - 1.80i)T - 53iT^{2} \)
59 \( 1 + 4.13T + 59T^{2} \)
61 \( 1 + 13.6iT - 61T^{2} \)
67 \( 1 + (2.59 + 2.59i)T + 67iT^{2} \)
71 \( 1 + (6.09 - 6.09i)T - 71iT^{2} \)
73 \( 1 + 2.91iT - 73T^{2} \)
79 \( 1 + (8.34 - 8.34i)T - 79iT^{2} \)
83 \( 1 + 5.43T + 83T^{2} \)
89 \( 1 + (-9.82 + 9.82i)T - 89iT^{2} \)
97 \( 1 + (8.27 + 8.27i)T + 97iT^{2} \)
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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

−11.83579958084339329056906427215, −10.83349689111099664117087264153, −10.22186625725823605958589086446, −8.558714969741929201742087441331, −7.86608693365470484030183461119, −7.03057346480867920477505396907, −6.25491368188164509719664161122, −4.73763837197526852063451075024, −2.90320402240168073593462893547, −2.01510646038394918789441324907, 1.83769832531165491558336836928, 3.24502417389854851534275655221, 4.29270282392531894118326243174, 5.81797519917575739408335420680, 6.82533511950613086278096737173, 8.404812798314538707806956140198, 8.705305202737492099180607076406, 10.22506575806979073695874839150, 10.56637642372351981890910240727, 11.97398758805705195740629131173

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