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Properties

Label 2-3120-195.122-c0-0-1
Degree $2$
Conductor $3120$
Sign $0.661 - 0.749i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s − i·7-s + 1.00i·9-s + (0.707 − 0.707i)11-s + i·13-s + 1.00i·15-s + (−0.707 − 0.707i)17-s + (0.707 − 0.707i)21-s + (0.707 − 0.707i)23-s + 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41·29-s + (−1 + i)31-s + 1.00·33-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s − i·7-s + 1.00i·9-s + (0.707 − 0.707i)11-s + i·13-s + 1.00i·15-s + (−0.707 − 0.707i)17-s + (0.707 − 0.707i)21-s + (0.707 − 0.707i)23-s + 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41·29-s + (−1 + i)31-s + 1.00·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (2657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ 0.661 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.872216045\)
\(L(\frac12)\) \(\approx\) \(1.872216045\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 - iT \)
good7 \( 1 + iT - T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
17 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + (1 - i)T - iT^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.139554290536881670195572452249, −8.419874842823164825497219186793, −7.34536696732913308665487707440, −6.81077560104794887054463774681, −6.09982901254210456000431346533, −4.86728290294784330134708857124, −4.28599258029621100418792347155, −3.35192045723604296819881186937, −2.65259731239866526549579341969, −1.48812304807916829593869462520, 1.25596237479601310941479133565, 2.11264555175771827286220582656, 2.88320933533929602893377955523, 4.02927014388250517208914702574, 5.01845898593493429838979689943, 5.89627847301230035949554088829, 6.40684584344730400109775307392, 7.37421179721116574833202649674, 8.156061090118880080135598675797, 8.862148565361649427130747078899

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