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Properties

Label 2-2340-13.10-c1-0-22
Degree $2$
Conductor $2340$
Sign $-0.943 + 0.331i$
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  − i·5-s + (0.346 + 0.199i)7-s + (−1.5 + 0.866i)11-s + (−0.619 + 3.55i)13-s + (−0.346 + 0.599i)17-s + (−4.65 − 2.68i)19-s + (0.0535 + 0.0927i)23-s − 25-s + (−2.45 − 4.24i)29-s − 7.86i·31-s + (0.199 − 0.346i)35-s + (1.96 − 1.13i)37-s + (−6.69 + 3.86i)41-s + (3.00 − 5.20i)43-s − 3.46i·47-s + ⋯
L(s)  = 1  − 0.447i·5-s + (0.130 + 0.0755i)7-s + (−0.452 + 0.261i)11-s + (−0.171 + 0.985i)13-s + (−0.0839 + 0.145i)17-s + (−1.06 − 0.616i)19-s + (0.0111 + 0.0193i)23-s − 0.200·25-s + (−0.455 − 0.788i)29-s − 1.41i·31-s + (0.0337 − 0.0585i)35-s + (0.322 − 0.186i)37-s + (−1.04 + 0.603i)41-s + (0.458 − 0.793i)43-s − 0.505i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.943 + 0.331i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.943 + 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3582257104\)
\(L(\frac12)\) \(\approx\) \(0.3582257104\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
13 \( 1 + (0.619 - 3.55i)T \)
good7 \( 1 + (-0.346 - 0.199i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.346 - 0.599i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.65 + 2.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0535 - 0.0927i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.45 + 4.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.86iT - 31T^{2} \)
37 \( 1 + (-1.96 + 1.13i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.69 - 3.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.00 + 5.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + (-6.30 - 3.63i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.34 - 7.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.15 - 0.664i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.35 - 1.93i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 14.0iT - 83T^{2} \)
89 \( 1 + (0.300 - 0.173i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.66 + 4.42i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.631243384288839367119327624149, −7.974811329836274572686595316743, −7.09709990712282333207255376621, −6.33536605336758798695449209256, −5.45379581134603223180048367835, −4.54413166016346070894995478104, −3.98252678136914749530292606075, −2.56513163685088595305504760069, −1.75841458482865069786704794768, −0.11454823646627931571634013611, 1.53861732611208444935974935889, 2.77702922276735699628491166163, 3.47090161290081223364337643667, 4.61406934228335682472831529454, 5.41548547974058376135594214132, 6.24709405872591444601627550403, 7.01553889231097465935857465336, 7.908693071951915281252880897732, 8.387427367464410495416007436704, 9.351578466735721110223159086823

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