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Properties

Label 2-4140-15.8-c1-0-31
Degree $2$
Conductor $4140$
Sign $0.981 + 0.191i$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
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  + (1.13 + 1.92i)5-s + (2.43 − 2.43i)7-s − 5.50i·11-s + (1.03 + 1.03i)13-s + (−0.543 − 0.543i)17-s + 5.88i·19-s + (−0.707 + 0.707i)23-s + (−2.43 + 4.36i)25-s + 7.99·29-s + 3.34·31-s + (7.43 + 1.93i)35-s + (−3.41 + 3.41i)37-s − 4.57i·41-s + (−1.70 − 1.70i)43-s + (7.40 + 7.40i)47-s + ⋯
L(s)  = 1  + (0.506 + 0.862i)5-s + (0.918 − 0.918i)7-s − 1.65i·11-s + (0.286 + 0.286i)13-s + (−0.131 − 0.131i)17-s + 1.34i·19-s + (−0.147 + 0.147i)23-s + (−0.487 + 0.872i)25-s + 1.48·29-s + 0.601·31-s + (1.25 + 0.327i)35-s + (−0.561 + 0.561i)37-s − 0.714i·41-s + (−0.259 − 0.259i)43-s + (1.08 + 1.08i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.981 + 0.191i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ 0.981 + 0.191i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.501360219\)
\(L(\frac12)\) \(\approx\) \(2.501360219\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.13 - 1.92i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-2.43 + 2.43i)T - 7iT^{2} \)
11 \( 1 + 5.50iT - 11T^{2} \)
13 \( 1 + (-1.03 - 1.03i)T + 13iT^{2} \)
17 \( 1 + (0.543 + 0.543i)T + 17iT^{2} \)
19 \( 1 - 5.88iT - 19T^{2} \)
29 \( 1 - 7.99T + 29T^{2} \)
31 \( 1 - 3.34T + 31T^{2} \)
37 \( 1 + (3.41 - 3.41i)T - 37iT^{2} \)
41 \( 1 + 4.57iT - 41T^{2} \)
43 \( 1 + (1.70 + 1.70i)T + 43iT^{2} \)
47 \( 1 + (-7.40 - 7.40i)T + 47iT^{2} \)
53 \( 1 + (-4.73 + 4.73i)T - 53iT^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + (-3.06 + 3.06i)T - 67iT^{2} \)
71 \( 1 + 10.1iT - 71T^{2} \)
73 \( 1 + (2.22 + 2.22i)T + 73iT^{2} \)
79 \( 1 - 6.61iT - 79T^{2} \)
83 \( 1 + (-3.00 + 3.00i)T - 83iT^{2} \)
89 \( 1 - 2.16T + 89T^{2} \)
97 \( 1 + (-11.3 + 11.3i)T - 97iT^{2} \)
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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

−8.303150993551798308304594835697, −7.73702336875079804482404854399, −6.88800459914180145000233773010, −6.17520833744494627241306562559, −5.61003384683909570443483164937, −4.57407751229268616051110815405, −3.69340699044053088372480284380, −3.03164963291839028886735212173, −1.85589554970352433331107715899, −0.886186218359195706703225424654, 1.01249531096815692436585551140, 2.06631965186609609082232074320, 2.59205439954386545751725598791, 4.18047901331786398781441517564, 4.84562833160380021518591808035, 5.22396248549993278416815556549, 6.17573736257342398891642295291, 6.98695353404364283434744809064, 7.81466246340534020160286909401, 8.709958332855464098464836267299

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