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Properties

Label 2-2340-39.20-c1-0-17
Degree $2$
Conductor $2340$
Sign $0.537 + 0.843i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
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  + (0.707 − 0.707i)5-s + (1.13 + 0.303i)7-s + (5.67 − 1.52i)11-s + (1.18 − 3.40i)13-s + (−1.54 − 2.67i)17-s + (0.197 − 0.737i)19-s + (−0.483 + 0.837i)23-s − 1.00i·25-s + (1.50 + 0.871i)29-s + (−6.73 − 6.73i)31-s + (1.01 − 0.586i)35-s + (1.40 + 5.23i)37-s + (−1.24 − 4.63i)41-s + (−0.193 + 0.111i)43-s + (5.91 + 5.91i)47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s + (0.428 + 0.114i)7-s + (1.71 − 0.458i)11-s + (0.329 − 0.944i)13-s + (−0.374 − 0.649i)17-s + (0.0453 − 0.169i)19-s + (−0.100 + 0.174i)23-s − 0.200i·25-s + (0.280 + 0.161i)29-s + (−1.20 − 1.20i)31-s + (0.171 − 0.0991i)35-s + (0.230 + 0.860i)37-s + (−0.194 − 0.724i)41-s + (−0.0295 + 0.0170i)43-s + (0.862 + 0.862i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.537 + 0.843i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (2321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.537 + 0.843i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.163434276\)
\(L(\frac12)\) \(\approx\) \(2.163434276\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-1.18 + 3.40i)T \)
good7 \( 1 + (-1.13 - 0.303i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-5.67 + 1.52i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.54 + 2.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.197 + 0.737i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.483 - 0.837i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.50 - 0.871i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.73 + 6.73i)T + 31iT^{2} \)
37 \( 1 + (-1.40 - 5.23i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.24 + 4.63i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (0.193 - 0.111i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.91 - 5.91i)T + 47iT^{2} \)
53 \( 1 - 6.79iT - 53T^{2} \)
59 \( 1 + (2.39 - 8.93i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.59 + 9.68i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.68 + 0.451i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.138 + 0.0372i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.76 + 1.76i)T - 73iT^{2} \)
79 \( 1 - 3.45T + 79T^{2} \)
83 \( 1 + (-10.0 + 10.0i)T - 83iT^{2} \)
89 \( 1 + (-8.76 + 2.34i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.444 + 1.65i)T + (-84.0 - 48.5i)T^{2} \)
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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

−9.117950577695612043112375952054, −8.147956114580442286037773499709, −7.38423290146145467557054742346, −6.39453287515261913087258801402, −5.81853867066998727565153522234, −4.89523882864629045897510947018, −4.01929091283391689329044900847, −3.10660896035028381220041109010, −1.84314433584706791770062490828, −0.805214666704091955723834497860, 1.39022309704755204713995934391, 2.07908619138830751293660945149, 3.58567977041762916280419209630, 4.16065655515400257212850538940, 5.10643581756059025726903738968, 6.26317825287780086204986693926, 6.65917266484242450849215728789, 7.44899466516659804070560270812, 8.520610046331551355979040940022, 9.107539133284510753434282799083

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