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Properties

Label 2-240-15.2-c1-0-2
Degree $2$
Conductor $240$
Sign $0.998 - 0.0618i$
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 − 1.41i)3-s + (1 + 2i)5-s + (2.41 + 2.41i)7-s + (−1.00 + 2.82i)9-s − 0.828i·11-s + (3.82 − 3.82i)13-s + (1.82 − 3.41i)15-s + (1.82 − 1.82i)17-s + 0.828i·19-s + (1 − 5.82i)21-s + (4.41 + 4.41i)23-s + (−3 + 4i)25-s + (5.00 − 1.41i)27-s − 3.65·29-s − 5.65·31-s + ⋯
L(s)  = 1  + (−0.577 − 0.816i)3-s + (0.447 + 0.894i)5-s + (0.912 + 0.912i)7-s + (−0.333 + 0.942i)9-s − 0.249i·11-s + (1.06 − 1.06i)13-s + (0.472 − 0.881i)15-s + (0.443 − 0.443i)17-s + 0.190i·19-s + (0.218 − 1.27i)21-s + (0.920 + 0.920i)23-s + (−0.600 + 0.800i)25-s + (0.962 − 0.272i)27-s − 0.679·29-s − 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.998 - 0.0618i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.998 - 0.0618i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22178 + 0.0378294i\)
\(L(\frac12)\) \(\approx\) \(1.22178 + 0.0378294i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1 + 1.41i)T \)
5 \( 1 + (-1 - 2i)T \)
good7 \( 1 + (-2.41 - 2.41i)T + 7iT^{2} \)
11 \( 1 + 0.828iT - 11T^{2} \)
13 \( 1 + (-3.82 + 3.82i)T - 13iT^{2} \)
17 \( 1 + (-1.82 + 1.82i)T - 17iT^{2} \)
19 \( 1 - 0.828iT - 19T^{2} \)
23 \( 1 + (-4.41 - 4.41i)T + 23iT^{2} \)
29 \( 1 + 3.65T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (5.82 + 5.82i)T + 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-0.414 + 0.414i)T - 43iT^{2} \)
47 \( 1 + (-3.58 + 3.58i)T - 47iT^{2} \)
53 \( 1 + (-3 - 3i)T + 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 0.343T + 61T^{2} \)
67 \( 1 + (10.0 + 10.0i)T + 67iT^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (4.65 - 4.65i)T - 73iT^{2} \)
79 \( 1 + 0.828iT - 79T^{2} \)
83 \( 1 + (-3.24 - 3.24i)T + 83iT^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (-1 - i)T + 97iT^{2} \)
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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

−12.02551409842224077357113619242, −11.14026401156295071045078433956, −10.66155193209759286890558472485, −9.133970728922215987650541239219, −8.011343554578811119457739131543, −7.15492821411022272513245463474, −5.80758421365607720424360520109, −5.42126457635478722499873033198, −3.14442093606505853477105600409, −1.69741552337889595398284927044, 1.36317808131470439466068699911, 3.93474741741486724813910761269, 4.70765864556558544360177137702, 5.74728602771783253441411456528, 7.00409794465703421322718669228, 8.501945218545773747384785900997, 9.196959688752711534374351639167, 10.36351934386180684221868601840, 11.04041055988904653411241720105, 11.95411406716806429500872805894

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