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Properties

Label 2-185-185.103-c1-0-1
Degree $2$
Conductor $185$
Sign $-0.851 - 0.524i$
Analytic cond. $1.47723$
Root an. cond. $1.21541$
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  + (1.35 + 2.34i)2-s + (−0.117 − 0.438i)3-s + (−2.65 + 4.60i)4-s + (−2.04 − 0.908i)5-s + (0.868 − 0.868i)6-s + (0.933 + 3.48i)7-s − 8.97·8-s + (2.41 − 1.39i)9-s + (−0.633 − 6.01i)10-s + 3.16i·11-s + (2.33 + 0.624i)12-s + (1.92 − 3.33i)13-s + (−6.90 + 6.90i)14-s + (−0.158 + 1.00i)15-s + (−6.81 − 11.8i)16-s + (1.94 − 1.12i)17-s + ⋯
L(s)  = 1  + (0.956 + 1.65i)2-s + (−0.0678 − 0.253i)3-s + (−1.32 + 2.30i)4-s + (−0.913 − 0.406i)5-s + (0.354 − 0.354i)6-s + (0.352 + 1.31i)7-s − 3.17·8-s + (0.806 − 0.465i)9-s + (−0.200 − 1.90i)10-s + 0.953i·11-s + (0.673 + 0.180i)12-s + (0.534 − 0.925i)13-s + (−1.84 + 1.84i)14-s + (−0.0409 + 0.258i)15-s + (−1.70 − 2.95i)16-s + (0.472 − 0.272i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.524i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.851 - 0.524i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.851 - 0.524i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421945 + 1.48786i\)
\(L(\frac12)\) \(\approx\) \(0.421945 + 1.48786i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.04 + 0.908i)T \)
37 \( 1 + (6.02 + 0.825i)T \)
good2 \( 1 + (-1.35 - 2.34i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.117 + 0.438i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.933 - 3.48i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 3.16iT - 11T^{2} \)
13 \( 1 + (-1.92 + 3.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.94 + 1.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.27 + 1.41i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 3.05T + 23T^{2} \)
29 \( 1 + (-3.71 + 3.71i)T - 29iT^{2} \)
31 \( 1 + (-0.870 - 0.870i)T + 31iT^{2} \)
41 \( 1 + (6.48 + 3.74i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + 2.70T + 43T^{2} \)
47 \( 1 + (5.24 - 5.24i)T - 47iT^{2} \)
53 \( 1 + (1.76 - 6.58i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.87 + 6.99i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.84 + 1.02i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-10.3 + 2.76i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.11 + 3.65i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.58 + 2.58i)T - 73iT^{2} \)
79 \( 1 + (7.11 - 1.90i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-1.03 + 3.86i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (3.50 + 0.940i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−13.01017492515853047484293678661, −12.25421552590972136149777290578, −11.87826050635118094159411979905, −9.562203255266332249226360747364, −8.413892021243444486353298823997, −7.73765954914698658723635741482, −6.78369282191598388798869795504, −5.53090676354549323617230131695, −4.73069676569817147658766918719, −3.41884393630429701122508731023, 1.30447149252308912599718306527, 3.45069381955357433245455914391, 4.01836512644743790074883608260, 5.10174770264341688059218472917, 6.82364887978198802975292006698, 8.301279186322835326331302397105, 9.945201645062304042391431156652, 10.51559172318818667521091072960, 11.34531043418363864420732799865, 11.90212209067252647892599495773

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