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Properties

Label 2-920-115.102-c1-0-18
Degree $2$
Conductor $920$
Sign $0.372 + 0.928i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  + (−1.16 + 0.252i)3-s + (−2.21 + 0.333i)5-s + (−3.31 + 1.80i)7-s + (−1.44 + 0.658i)9-s + (2.85 + 2.47i)11-s + (0.632 − 1.15i)13-s + (2.48 − 0.946i)15-s + (−3.12 − 2.34i)17-s + (−0.174 + 1.21i)19-s + (3.39 − 2.93i)21-s + (3.10 − 3.65i)23-s + (4.77 − 1.47i)25-s + (4.36 − 3.26i)27-s + (−1.68 + 0.241i)29-s + (2.63 − 1.69i)31-s + ⋯
L(s)  = 1  + (−0.670 + 0.145i)3-s + (−0.988 + 0.149i)5-s + (−1.25 + 0.683i)7-s + (−0.480 + 0.219i)9-s + (0.860 + 0.745i)11-s + (0.175 − 0.321i)13-s + (0.641 − 0.244i)15-s + (−0.758 − 0.568i)17-s + (−0.0401 + 0.279i)19-s + (0.739 − 0.641i)21-s + (0.646 − 0.762i)23-s + (0.955 − 0.294i)25-s + (0.840 − 0.628i)27-s + (−0.312 + 0.0449i)29-s + (0.472 − 0.303i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.372 + 0.928i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.372 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.341088 - 0.230653i\)
\(L(\frac12)\) \(\approx\) \(0.341088 - 0.230653i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.21 - 0.333i)T \)
23 \( 1 + (-3.10 + 3.65i)T \)
good3 \( 1 + (1.16 - 0.252i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (3.31 - 1.80i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (-2.85 - 2.47i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-0.632 + 1.15i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (3.12 + 2.34i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (0.174 - 1.21i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (1.68 - 0.241i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (-2.63 + 1.69i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (6.09 + 2.27i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (-0.616 + 1.34i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (2.02 + 9.30i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (-5.17 - 5.17i)T + 47iT^{2} \)
53 \( 1 + (6.61 + 12.1i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (-2.63 - 8.98i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (0.888 + 1.38i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-3.61 - 0.258i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (1.30 + 1.50i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-7.36 - 9.83i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (3.37 - 0.990i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (1.51 - 4.05i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (-12.2 - 7.89i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (3.82 + 10.2i)T + (-73.3 + 63.5i)T^{2} \)
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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

−9.975414199933698268765153674286, −9.044023509637328919127678301249, −8.412366630307809135403555725663, −7.08982335119402689466426686612, −6.58662278165182865077640684146, −5.60717049715982903164372410882, −4.59433371578260086107907429753, −3.56645170153189950519686690584, −2.52145523679826344173814190814, −0.27712174658552493058834870511, 0.957444794191246036096725479755, 3.20334629168048407869717564131, 3.78991517883581967759549903770, 4.92090233683972084939280787169, 6.30482629148976256538007222135, 6.55471428003629994033796928050, 7.56959957188509687478823006683, 8.740296724620346956322677391527, 9.223866718285513530988651044458, 10.45216093771385411707813801168

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