Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 4-s − 5-s − 2·6-s − 2·7-s − 3·8-s + 9-s − 10-s + 2·12-s − 2·13-s − 2·14-s + 2·15-s − 16-s + 2·17-s + 18-s + 2·19-s + 20-s + 4·21-s − 8·23-s + 6·24-s + 25-s − 2·26-s + 4·27-s + 2·28-s + 2·29-s + 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s − 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s − 0.554·13-s − 0.534·14-s + 0.516·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.458·19-s + 0.223·20-s + 0.872·21-s − 1.66·23-s + 1.22·24-s + 1/5·25-s − 0.392·26-s + 0.769·27-s + 0.377·28-s + 0.371·29-s + 0.365·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-1$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad5 \( 1 + T \)
37 \( 1 + T \)
good2 \( 1 - T + p T^{2} \) 1.2.ab
3 \( 1 + 2 T + p T^{2} \) 1.3.c
7 \( 1 + 2 T + p T^{2} \) 1.7.c
11 \( 1 + p T^{2} \) 1.11.a
13 \( 1 + 2 T + p T^{2} \) 1.13.c
17 \( 1 - 2 T + p T^{2} \) 1.17.ac
19 \( 1 - 2 T + p T^{2} \) 1.19.ac
23 \( 1 + 8 T + p T^{2} \) 1.23.i
29 \( 1 - 2 T + p T^{2} \) 1.29.ac
31 \( 1 + 6 T + p T^{2} \) 1.31.g
41 \( 1 - 10 T + p T^{2} \) 1.41.ak
43 \( 1 + 4 T + p T^{2} \) 1.43.e
47 \( 1 + 10 T + p T^{2} \) 1.47.k
53 \( 1 + 6 T + p T^{2} \) 1.53.g
59 \( 1 + 6 T + p T^{2} \) 1.59.g
61 \( 1 - 2 T + p T^{2} \) 1.61.ac
67 \( 1 + 14 T + p T^{2} \) 1.67.o
71 \( 1 + p T^{2} \) 1.71.a
73 \( 1 - 2 T + p T^{2} \) 1.73.ac
79 \( 1 + 6 T + p T^{2} \) 1.79.g
83 \( 1 - 18 T + p T^{2} \) 1.83.as
89 \( 1 - 2 T + p T^{2} \) 1.89.ac
97 \( 1 + 10 T + p T^{2} \) 1.97.k
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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

−12.22791693199162869617149037713, −11.49667671710560792633585266068, −10.23148599364934618296079461514, −9.307883159759705212702301295277, −7.87513819180148495071092374749, −6.43570637045934690326444081037, −5.61819870783945714975164299241, −4.57786943494299050334574286365, −3.30715590158476991369802502567, 0, 3.30715590158476991369802502567, 4.57786943494299050334574286365, 5.61819870783945714975164299241, 6.43570637045934690326444081037, 7.87513819180148495071092374749, 9.307883159759705212702301295277, 10.23148599364934618296079461514, 11.49667671710560792633585266068, 12.22791693199162869617149037713

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