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Properties

Label 2-240-80.43-c1-0-18
Degree $2$
Conductor $240$
Sign $0.330 + 0.943i$
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  + (0.873 − 1.11i)2-s + i·3-s + (−0.472 − 1.94i)4-s + (1.54 − 1.61i)5-s + (1.11 + 0.873i)6-s + (0.143 − 0.143i)7-s + (−2.57 − 1.17i)8-s − 9-s + (−0.441 − 3.13i)10-s + (0.749 − 0.749i)11-s + (1.94 − 0.472i)12-s + 3.29·13-s + (−0.0342 − 0.285i)14-s + (1.61 + 1.54i)15-s + (−3.55 + 1.83i)16-s + (1.35 − 1.35i)17-s + ⋯
L(s)  = 1  + (0.617 − 0.786i)2-s + 0.577i·3-s + (−0.236 − 0.971i)4-s + (0.692 − 0.721i)5-s + (0.453 + 0.356i)6-s + (0.0543 − 0.0543i)7-s + (−0.910 − 0.414i)8-s − 0.333·9-s + (−0.139 − 0.990i)10-s + (0.225 − 0.225i)11-s + (0.560 − 0.136i)12-s + 0.912·13-s + (−0.00915 − 0.0763i)14-s + (0.416 + 0.399i)15-s + (−0.888 + 0.459i)16-s + (0.329 − 0.329i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.330 + 0.943i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.330 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44531 - 1.02495i\)
\(L(\frac12)\) \(\approx\) \(1.44531 - 1.02495i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.873 + 1.11i)T \)
3 \( 1 - iT \)
5 \( 1 + (-1.54 + 1.61i)T \)
good7 \( 1 + (-0.143 + 0.143i)T - 7iT^{2} \)
11 \( 1 + (-0.749 + 0.749i)T - 11iT^{2} \)
13 \( 1 - 3.29T + 13T^{2} \)
17 \( 1 + (-1.35 + 1.35i)T - 17iT^{2} \)
19 \( 1 + (4.25 - 4.25i)T - 19iT^{2} \)
23 \( 1 + (-0.837 - 0.837i)T + 23iT^{2} \)
29 \( 1 + (-2.77 - 2.77i)T + 29iT^{2} \)
31 \( 1 - 6.60iT - 31T^{2} \)
37 \( 1 + 10.0T + 37T^{2} \)
41 \( 1 - 1.72iT - 41T^{2} \)
43 \( 1 + 4.17T + 43T^{2} \)
47 \( 1 + (-8.54 - 8.54i)T + 47iT^{2} \)
53 \( 1 + 5.05iT - 53T^{2} \)
59 \( 1 + (3.08 + 3.08i)T + 59iT^{2} \)
61 \( 1 + (5.00 - 5.00i)T - 61iT^{2} \)
67 \( 1 + 4.26T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + (-11.6 + 11.6i)T - 73iT^{2} \)
79 \( 1 + 9.95T + 79T^{2} \)
83 \( 1 + 10.0iT - 83T^{2} \)
89 \( 1 + 5.76T + 89T^{2} \)
97 \( 1 + (-11.7 + 11.7i)T - 97iT^{2} \)
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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

−12.09133699795710962717038234128, −10.85886492448757168128585135005, −10.26039048515287994914806082461, −9.198382134441720658659564471867, −8.488296125776681384137347008115, −6.40327419159478601850465700295, −5.49397688942771325594409485989, −4.48569543645531444963147732377, −3.29367017900049301165823669005, −1.52826763402621299924179740480, 2.35632412656323122187235916604, 3.80602962614268793695364862115, 5.37685946810423193799986158143, 6.38840342098430119085586724232, 6.96968289549308023583451594644, 8.207828374674153831748677568110, 9.123931200162191664067756479181, 10.53376874270399533332623578421, 11.55645137554907778703701418611, 12.59142929115318212830001664388

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