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Properties

Label 2-3736-3736.291-c0-0-0
Degree $2$
Conductor $3736$
Sign $-0.628 + 0.777i$
Analytic cond. $1.86450$
Root an. cond. $1.36546$
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  + (0.829 − 0.559i)2-s + (0.147 + 0.00998i)3-s + (0.374 − 0.927i)4-s + (0.128 − 0.0743i)6-s + (−0.207 − 0.978i)8-s + (−0.969 − 0.131i)9-s + (−0.211 − 0.503i)11-s + (0.0646 − 0.133i)12-s + (−0.718 − 0.695i)16-s + (−0.778 − 1.78i)17-s + (−0.877 + 0.432i)18-s + (1.11 − 0.291i)19-s + (−0.456 − 0.299i)22-s + (−0.0209 − 0.146i)24-s + (−0.0471 + 0.998i)25-s + ⋯
L(s)  = 1  + (0.829 − 0.559i)2-s + (0.147 + 0.00998i)3-s + (0.374 − 0.927i)4-s + (0.128 − 0.0743i)6-s + (−0.207 − 0.978i)8-s + (−0.969 − 0.131i)9-s + (−0.211 − 0.503i)11-s + (0.0646 − 0.133i)12-s + (−0.718 − 0.695i)16-s + (−0.778 − 1.78i)17-s + (−0.877 + 0.432i)18-s + (1.11 − 0.291i)19-s + (−0.456 − 0.299i)22-s + (−0.0209 − 0.146i)24-s + (−0.0471 + 0.998i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3736\)    =    \(2^{3} \cdot 467\)
Sign: $-0.628 + 0.777i$
Analytic conductor: \(1.86450\)
Root analytic conductor: \(1.36546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3736} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3736,\ (\ :0),\ -0.628 + 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.769513058\)
\(L(\frac12)\) \(\approx\) \(1.769513058\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.829 + 0.559i)T \)
467 \( 1 + (-0.843 - 0.536i)T \)
good3 \( 1 + (-0.147 - 0.00998i)T + (0.990 + 0.134i)T^{2} \)
5 \( 1 + (0.0471 - 0.998i)T^{2} \)
7 \( 1 + (-0.948 - 0.317i)T^{2} \)
11 \( 1 + (0.211 + 0.503i)T + (-0.699 + 0.714i)T^{2} \)
13 \( 1 + (0.507 - 0.861i)T^{2} \)
17 \( 1 + (0.778 + 1.78i)T + (-0.680 + 0.732i)T^{2} \)
19 \( 1 + (-1.11 + 0.291i)T + (0.871 - 0.490i)T^{2} \)
23 \( 1 + (-0.472 - 0.881i)T^{2} \)
29 \( 1 + (-0.167 + 0.985i)T^{2} \)
31 \( 1 + (0.979 + 0.200i)T^{2} \)
37 \( 1 + (0.984 + 0.174i)T^{2} \)
41 \( 1 + (-0.0619 + 0.0265i)T + (0.690 - 0.723i)T^{2} \)
43 \( 1 + (0.249 + 1.93i)T + (-0.967 + 0.253i)T^{2} \)
47 \( 1 + (-0.956 + 0.292i)T^{2} \)
53 \( 1 + (-0.986 - 0.161i)T^{2} \)
59 \( 1 + (1.38 - 0.854i)T + (0.448 - 0.893i)T^{2} \)
61 \( 1 + (0.639 + 0.768i)T^{2} \)
67 \( 1 + (-0.472 + 0.444i)T + (0.0606 - 0.998i)T^{2} \)
71 \( 1 + (-0.970 - 0.240i)T^{2} \)
73 \( 1 + (-1.37 + 0.244i)T + (0.939 - 0.343i)T^{2} \)
79 \( 1 + (-0.194 - 0.980i)T^{2} \)
83 \( 1 + (1.15 - 1.59i)T + (-0.311 - 0.950i)T^{2} \)
89 \( 1 + (-1.20 + 0.281i)T + (0.896 - 0.442i)T^{2} \)
97 \( 1 + (-0.388 + 1.24i)T + (-0.821 - 0.570i)T^{2} \)
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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

−8.658983660640543325329934050263, −7.44611085409762303028919246759, −6.95695380823309120947766613717, −5.89806094984317215464050650257, −5.35326158863400882765993064349, −4.70794544815464664716329403448, −3.54951002699249185914868735190, −2.97769782379702693674730359240, −2.20201298833628890412280912960, −0.71230454852018310846335486587, 1.88450075204575503591808568428, 2.78475408190646730942559062949, 3.63573938805058887125955664500, 4.45563873051116742080456770450, 5.22454551938453449627569707553, 6.07264873490241723649720635283, 6.47329876648382403529598257061, 7.54681356520067697496320467364, 8.099507053386259452057346102478, 8.645483141819981097106715049027

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