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Properties

Label 2-240-16.13-c1-0-12
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 − 0.828i·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 + (−3.41 − 3.41i)13-s − 1.17i·14-s − 1.00·15-s + 4.00·16-s + 2.58·17-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 − 0.313i·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 + (−0.946 − 0.946i)13-s − 0.313i·14-s − 0.258·15-s + 1.00·16-s + 0.627·17-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

\(L(1)\) \(\approx\) \(2.26242 + 0.450023i\)
\(L(\frac12)\) \(\approx\) \(2.26242 + 0.450023i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.828iT - 7T^{2} \)
11 \( 1 + (1.41 - 1.41i)T - 11iT^{2} \)
13 \( 1 + (3.41 + 3.41i)T + 13iT^{2} \)
17 \( 1 - 2.58T + 17T^{2} \)
19 \( 1 + (1.82 + 1.82i)T + 19iT^{2} \)
23 \( 1 - 2.58iT - 23T^{2} \)
29 \( 1 + (3.41 + 3.41i)T + 29iT^{2} \)
31 \( 1 + 7.65T + 31T^{2} \)
37 \( 1 + (-7.41 + 7.41i)T - 37iT^{2} \)
41 \( 1 - 0.828iT - 41T^{2} \)
43 \( 1 + (7.65 - 7.65i)T - 43iT^{2} \)
47 \( 1 - 7.07T + 47T^{2} \)
53 \( 1 + (4 - 4i)T - 53iT^{2} \)
59 \( 1 + (4.58 - 4.58i)T - 59iT^{2} \)
61 \( 1 + (7.48 + 7.48i)T + 61iT^{2} \)
67 \( 1 + (-3.65 - 3.65i)T + 67iT^{2} \)
71 \( 1 - 8iT - 71T^{2} \)
73 \( 1 + 8.82iT - 73T^{2} \)
79 \( 1 + 2.34T + 79T^{2} \)
83 \( 1 + (-11.0 - 11.0i)T + 83iT^{2} \)
89 \( 1 - 7.65iT - 89T^{2} \)
97 \( 1 - 9.31T + 97T^{2} \)
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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

−12.42390531037018050077613964319, −11.23286755387120111925201101276, −10.47545115215708421162356109563, −9.536099497765248080858440727793, −7.79146720642455903373203485711, −7.36295717829129518171451208565, −5.80751657087769581442204801696, −4.74800640910058714990889552478, −3.61186868636726762996567762519, −2.46334135533661805150984816510, 2.03243702539398947609941968412, 3.40186648805927477768863678636, 4.68762482665753680991487219528, 5.81872848539203951178800138263, 7.02505039690791042311385966011, 7.88626342527376309297885336020, 9.037754804412013554548740635878, 10.34078929381706374328755211747, 11.50793460805887932021523446783, 12.26683938206035048402615638912

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