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Properties

Label 2-240-16.13-c1-0-2
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: $\chi_{240} (61, \cdot )$
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

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

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