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Properties

Label 2-240-16.5-c1-0-8
Degree $2$
Conductor $240$
Sign $-0.0985 + 0.995i$
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  + (−1.34 − 0.443i)2-s + (0.707 − 0.707i)3-s + (1.60 + 1.19i)4-s + (−0.707 − 0.707i)5-s + (−1.26 + 0.635i)6-s − 1.41i·7-s + (−1.62 − 2.31i)8-s − 1.00i·9-s + (0.635 + 1.26i)10-s + (−0.526 − 0.526i)11-s + (1.97 − 0.292i)12-s + (3.68 − 3.68i)13-s + (−0.627 + 1.89i)14-s − 1.00·15-s + (1.15 + 3.82i)16-s − 1.57·17-s + ⋯
L(s)  = 1  + (−0.949 − 0.313i)2-s + (0.408 − 0.408i)3-s + (0.803 + 0.595i)4-s + (−0.316 − 0.316i)5-s + (−0.515 + 0.259i)6-s − 0.534i·7-s + (−0.575 − 0.817i)8-s − 0.333i·9-s + (0.201 + 0.399i)10-s + (−0.158 − 0.158i)11-s + (0.571 − 0.0845i)12-s + (1.02 − 1.02i)13-s + (−0.167 + 0.507i)14-s − 0.258·15-s + (0.289 + 0.957i)16-s − 0.381·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0985 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0985 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.0985 + 0.995i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.0985 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571769 - 0.631214i\)
\(L(\frac12)\) \(\approx\) \(0.571769 - 0.631214i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.443i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + 1.41iT - 7T^{2} \)
11 \( 1 + (0.526 + 0.526i)T + 11iT^{2} \)
13 \( 1 + (-3.68 + 3.68i)T - 13iT^{2} \)
17 \( 1 + 1.57T + 17T^{2} \)
19 \( 1 + (0.383 - 0.383i)T - 19iT^{2} \)
23 \( 1 + 6.42iT - 23T^{2} \)
29 \( 1 + (-4.38 + 4.38i)T - 29iT^{2} \)
31 \( 1 + 5.75T + 31T^{2} \)
37 \( 1 + (-1.91 - 1.91i)T + 37iT^{2} \)
41 \( 1 - 11.9iT - 41T^{2} \)
43 \( 1 + (1.12 + 1.12i)T + 43iT^{2} \)
47 \( 1 + 2.09T + 47T^{2} \)
53 \( 1 + (-9.55 - 9.55i)T + 53iT^{2} \)
59 \( 1 + (-4.61 - 4.61i)T + 59iT^{2} \)
61 \( 1 + (-4.53 + 4.53i)T - 61iT^{2} \)
67 \( 1 + (5.59 - 5.59i)T - 67iT^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + 3.33T + 79T^{2} \)
83 \( 1 + (-6.31 + 6.31i)T - 83iT^{2} \)
89 \( 1 + 5.42iT - 89T^{2} \)
97 \( 1 - 2.13T + 97T^{2} \)
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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

−11.73128886719556452076020668562, −10.79650279669864784244065927859, −10.00160808674725080801748942936, −8.671210589376024906091829946164, −8.215265849686620312685278592359, −7.18700801939516214327416014251, −6.07276005872492884438962880931, −4.07190412013409326047035914434, −2.75695660054304339171081918624, −0.939715458334371186043563379864, 2.03870166583402368665348333525, 3.63827564828868291805129527476, 5.34344829431777042091853278673, 6.58896791207080321388840048045, 7.56397588956708051599373927302, 8.745219967294651943743489748900, 9.181964299487641068089348828480, 10.39351242547116626949206562013, 11.18982326362922639674181301296, 12.01404287040010132131673543255

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