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Properties

Label 2-240-16.13-c1-0-3
Degree $2$
Conductor $240$
Sign $0.923 - 0.382i$
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  − 1.41·2-s + (−0.707 − 0.707i)3-s + 2.00·4-s + (0.707 − 0.707i)5-s + (1.00 + 1.00i)6-s + 4.82i·7-s − 2.82·8-s + 1.00i·9-s + (−1.00 + 1.00i)10-s + (1.41 − 1.41i)11-s + (−1.41 − 1.41i)12-s + (−0.585 − 0.585i)13-s − 6.82i·14-s − 1.00·15-s + 4.00·16-s + 5.41·17-s + ⋯
L(s)  = 1  − 1.00·2-s + (−0.408 − 0.408i)3-s + 1.00·4-s + (0.316 − 0.316i)5-s + (0.408 + 0.408i)6-s + 1.82i·7-s − 1.00·8-s + 0.333i·9-s + (−0.316 + 0.316i)10-s + (0.426 − 0.426i)11-s + (−0.408 − 0.408i)12-s + (−0.162 − 0.162i)13-s − 1.82i·14-s − 0.258·15-s + 1.00·16-s + 1.31·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.923 - 0.382i$
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: \((2,\ 240,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.756985 + 0.150573i\)
\(L(\frac12)\) \(\approx\) \(0.756985 + 0.150573i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - 4.82iT - 7T^{2} \)
11 \( 1 + (-1.41 + 1.41i)T - 11iT^{2} \)
13 \( 1 + (0.585 + 0.585i)T + 13iT^{2} \)
17 \( 1 - 5.41T + 17T^{2} \)
19 \( 1 + (-3.82 - 3.82i)T + 19iT^{2} \)
23 \( 1 - 5.41iT - 23T^{2} \)
29 \( 1 + (0.585 + 0.585i)T + 29iT^{2} \)
31 \( 1 - 3.65T + 31T^{2} \)
37 \( 1 + (-4.58 + 4.58i)T - 37iT^{2} \)
41 \( 1 + 4.82iT - 41T^{2} \)
43 \( 1 + (-3.65 + 3.65i)T - 43iT^{2} \)
47 \( 1 + 7.07T + 47T^{2} \)
53 \( 1 + (4 - 4i)T - 53iT^{2} \)
59 \( 1 + (7.41 - 7.41i)T - 59iT^{2} \)
61 \( 1 + (-9.48 - 9.48i)T + 61iT^{2} \)
67 \( 1 + (7.65 + 7.65i)T + 67iT^{2} \)
71 \( 1 - 8iT - 71T^{2} \)
73 \( 1 + 3.17iT - 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 + (3.07 + 3.07i)T + 83iT^{2} \)
89 \( 1 + 3.65iT - 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−12.00997781861705480559707207733, −11.43731714470109308484997392451, −10.01559233344978921601479386970, −9.277720885508561348299154919692, −8.369291420000108117648009067782, −7.43383293131584678399427208720, −5.77809637420541643914590886778, −5.72441896845645920336441898945, −2.98803578492664588374700669090, −1.52976246734574735767727593731, 1.05088622142351709605559101227, 3.22890133187389066318010250320, 4.69401700176412843598085722384, 6.36409482137706916702830528825, 7.12144329946936273066980732850, 8.037661479463518117034484027640, 9.665941886234363463481343680363, 9.956953406184057709066909059903, 10.90709430855887201377894957272, 11.62994744710004685833285007554

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