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Properties

Label 2-185-185.68-c1-0-1
Degree $2$
Conductor $185$
Sign $-0.0158 - 0.999i$
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  − 2-s + (1 − i)3-s − 4-s + (−1 + 2i)5-s + (−1 + i)6-s + (−3 + 3i)7-s + 3·8-s + i·9-s + (1 − 2i)10-s + 2i·11-s + (−1 + i)12-s + 2·13-s + (3 − 3i)14-s + (1 + 3i)15-s − 16-s − 4i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.577 − 0.577i)3-s − 0.5·4-s + (−0.447 + 0.894i)5-s + (−0.408 + 0.408i)6-s + (−1.13 + 1.13i)7-s + 1.06·8-s + 0.333i·9-s + (0.316 − 0.632i)10-s + 0.603i·11-s + (−0.288 + 0.288i)12-s + 0.554·13-s + (0.801 − 0.801i)14-s + (0.258 + 0.774i)15-s − 0.250·16-s − 0.970i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0158 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0158 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.0158 - 0.999i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ -0.0158 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.416790 + 0.423470i\)
\(L(\frac12)\) \(\approx\) \(0.416790 + 0.423470i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1 - 2i)T \)
37 \( 1 + (1 + 6i)T \)
good2 \( 1 + T + 2T^{2} \)
3 \( 1 + (-1 + i)T - 3iT^{2} \)
7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + (-3 - 3i)T + 19iT^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + (7 - 7i)T - 29iT^{2} \)
31 \( 1 + (-3 - 3i)T + 31iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (-5 + 5i)T - 47iT^{2} \)
53 \( 1 + (3 + 3i)T + 53iT^{2} \)
59 \( 1 + (-7 - 7i)T + 59iT^{2} \)
61 \( 1 + (-1 - i)T + 61iT^{2} \)
67 \( 1 + (-3 - 3i)T + 67iT^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 + (-3 - 3i)T + 79iT^{2} \)
83 \( 1 + (5 + 5i)T + 83iT^{2} \)
89 \( 1 + (-5 + 5i)T - 89iT^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−12.83700396025873518719060313011, −11.99992072438835276443555855726, −10.62904111038075452920472992408, −9.680822224664481339355704274917, −8.862834497522444094829440373628, −7.80613740742579314166288228579, −7.04349833318338023985203545872, −5.59419844091067508860520241629, −3.67341265117217924705438134962, −2.29561026262457437442647611969, 0.65300052429205403887487345028, 3.72657409626789884374060522575, 4.18449371322501977599649225864, 6.06537701611764299515641395379, 7.68277854882150940198308918500, 8.465687753732322302234960146005, 9.481537583834271805720213323745, 9.886485680215773428715856678466, 11.08622767691944824133600431390, 12.53421607436119648825260226143

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