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Properties

Label 2-240-80.69-c1-0-11
Degree $2$
Conductor $240$
Sign $0.332 - 0.942i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
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  + (−0.345 + 1.37i)2-s + (0.707 + 0.707i)3-s + (−1.76 − 0.946i)4-s + (2.16 − 0.561i)5-s + (−1.21 + 0.725i)6-s + 4.51·7-s + (1.90 − 2.08i)8-s + 1.00i·9-s + (0.0233 + 3.16i)10-s + (−3.44 − 3.44i)11-s + (−0.576 − 1.91i)12-s + (−0.113 − 0.113i)13-s + (−1.55 + 6.19i)14-s + (1.92 + 1.13i)15-s + (2.20 + 3.33i)16-s + 5.03i·17-s + ⋯
L(s)  = 1  + (−0.244 + 0.969i)2-s + (0.408 + 0.408i)3-s + (−0.880 − 0.473i)4-s + (0.967 − 0.251i)5-s + (−0.495 + 0.296i)6-s + 1.70·7-s + (0.673 − 0.738i)8-s + 0.333i·9-s + (0.00739 + 0.999i)10-s + (−1.03 − 1.03i)11-s + (−0.166 − 0.552i)12-s + (−0.0315 − 0.0315i)13-s + (−0.416 + 1.65i)14-s + (0.497 + 0.292i)15-s + (0.551 + 0.833i)16-s + 1.22i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.332 - 0.942i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.332 - 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17417 + 0.830580i\)
\(L(\frac12)\) \(\approx\) \(1.17417 + 0.830580i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.345 - 1.37i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-2.16 + 0.561i)T \)
good7 \( 1 - 4.51T + 7T^{2} \)
11 \( 1 + (3.44 + 3.44i)T + 11iT^{2} \)
13 \( 1 + (0.113 + 0.113i)T + 13iT^{2} \)
17 \( 1 - 5.03iT - 17T^{2} \)
19 \( 1 + (0.992 - 0.992i)T - 19iT^{2} \)
23 \( 1 + 8.00T + 23T^{2} \)
29 \( 1 + (1.01 - 1.01i)T - 29iT^{2} \)
31 \( 1 + 6.42T + 31T^{2} \)
37 \( 1 + (-1.63 + 1.63i)T - 37iT^{2} \)
41 \( 1 + 3.35iT - 41T^{2} \)
43 \( 1 + (-5.68 + 5.68i)T - 43iT^{2} \)
47 \( 1 - 9.10iT - 47T^{2} \)
53 \( 1 + (-3.27 + 3.27i)T - 53iT^{2} \)
59 \( 1 + (5.30 + 5.30i)T + 59iT^{2} \)
61 \( 1 + (5.87 - 5.87i)T - 61iT^{2} \)
67 \( 1 + (1.87 + 1.87i)T + 67iT^{2} \)
71 \( 1 - 0.635iT - 71T^{2} \)
73 \( 1 - 6.14T + 73T^{2} \)
79 \( 1 - 1.76T + 79T^{2} \)
83 \( 1 + (6.39 + 6.39i)T + 83iT^{2} \)
89 \( 1 + 0.579iT - 89T^{2} \)
97 \( 1 - 15.2iT - 97T^{2} \)
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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

−12.62156286081824562745386420238, −10.86281439065420238105505905098, −10.39234185815750166483014826949, −9.106455585700820136551847768930, −8.293117128000431669368872463213, −7.75078044932170662541858711854, −5.93240662354015849453623006027, −5.34777493393713894610196276124, −4.15305230235542712114433208727, −1.86048528269773877623943333593, 1.77173359189475306860394236513, 2.52325781925991007191473368350, 4.49428933579430945699175236953, 5.43284344587911516839595286214, 7.37357454477337787997957303812, 8.059041455906846429723122558246, 9.210122759224129909691472499085, 10.07801812137878045196389527237, 10.96846073361056267070850994023, 11.87736348340435985279860321337

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