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Properties

Label 2-185-185.174-c1-0-6
Degree $2$
Conductor $185$
Sign $0.0308 - 0.999i$
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  + (−0.398 + 0.230i)2-s + (2.88 + 1.66i)3-s + (−0.894 + 1.54i)4-s + (−2.18 − 0.462i)5-s − 1.53·6-s + (−0.0944 − 0.0545i)7-s − 1.74i·8-s + (4.04 + 7.00i)9-s + (0.978 − 0.319i)10-s + 2.27·11-s + (−5.15 + 2.97i)12-s + (−2.22 − 1.28i)13-s + 0.0501·14-s + (−5.54 − 4.97i)15-s + (−1.38 − 2.40i)16-s + (4.60 − 2.66i)17-s + ⋯
L(s)  = 1  + (−0.281 + 0.162i)2-s + (1.66 + 0.961i)3-s + (−0.447 + 0.774i)4-s + (−0.978 − 0.206i)5-s − 0.625·6-s + (−0.0356 − 0.0206i)7-s − 0.616i·8-s + (1.34 + 2.33i)9-s + (0.309 − 0.100i)10-s + 0.685·11-s + (−1.48 + 0.859i)12-s + (−0.616 − 0.356i)13-s + 0.0134·14-s + (−1.43 − 1.28i)15-s + (−0.346 − 0.600i)16-s + (1.11 − 0.645i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0308 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0308 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.0308 - 0.999i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.0308 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967359 + 0.937976i\)
\(L(\frac12)\) \(\approx\) \(0.967359 + 0.937976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.18 + 0.462i)T \)
37 \( 1 + (1.93 + 5.76i)T \)
good2 \( 1 + (0.398 - 0.230i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-2.88 - 1.66i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.0944 + 0.0545i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.27T + 11T^{2} \)
13 \( 1 + (2.22 + 1.28i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.60 + 2.66i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.75 - 3.03i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 - 1.44T + 29T^{2} \)
31 \( 1 - 1.87T + 31T^{2} \)
41 \( 1 + (-5.48 + 9.49i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4.53iT - 43T^{2} \)
47 \( 1 - 4.28iT - 47T^{2} \)
53 \( 1 + (5.36 - 3.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.03 + 5.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.57 - 6.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.76 + 5.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.53 - 2.66i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 7.89iT - 73T^{2} \)
79 \( 1 + (5.44 - 9.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.36 - 1.36i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.51 + 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.82iT - 97T^{2} \)
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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

−12.85013939323342320522406931346, −12.10435636720038277787724096539, −10.51097605928272928337584927715, −9.509680347428669858860470059894, −8.787127689890658688825155089690, −7.961896521620428092978096348319, −7.36671252191280197615093787034, −4.70284293100087977787130259374, −3.86845986900173139626842017763, −2.96128700014773525692965784577, 1.41822507327080423395215241494, 3.05774894952913688966997291199, 4.33898513912735278854404877187, 6.43586990542801089735705939050, 7.50765081085557545257651891492, 8.343872075693125945965436900393, 9.178193905032543711528403465952, 10.03523068634368687799631334225, 11.55893089758519767010608586813, 12.47534452967738473107036288933

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