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Properties

Label 2-185-185.104-c1-0-10
Degree $2$
Conductor $185$
Sign $0.409 + 0.912i$
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  + (−0.381 + 0.138i)2-s + (1.04 − 2.85i)3-s + (−1.40 + 1.17i)4-s + (2.23 + 0.0131i)5-s + 1.23i·6-s + (1.76 − 0.311i)7-s + (0.777 − 1.34i)8-s + (−4.79 − 4.02i)9-s + (−0.854 + 0.305i)10-s + (−0.401 + 0.696i)11-s + (1.91 + 5.24i)12-s + (0.737 − 0.619i)13-s + (−0.630 + 0.364i)14-s + (2.36 − 6.38i)15-s + (0.527 − 2.99i)16-s + (−4.96 − 4.16i)17-s + ⋯
L(s)  = 1  + (−0.269 + 0.0981i)2-s + (0.600 − 1.65i)3-s + (−0.703 + 0.589i)4-s + (0.999 + 0.00588i)5-s + 0.503i·6-s + (0.668 − 0.117i)7-s + (0.275 − 0.476i)8-s + (−1.59 − 1.34i)9-s + (−0.270 + 0.0965i)10-s + (−0.121 + 0.209i)11-s + (0.551 + 1.51i)12-s + (0.204 − 0.171i)13-s + (−0.168 + 0.0973i)14-s + (0.610 − 1.64i)15-s + (0.131 − 0.748i)16-s + (−1.20 − 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.409 + 0.912i$
Analytic conductor: \(1.47723\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 185,\ (\ :1/2),\ 0.409 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04185 - 0.674091i\)
\(L(\frac12)\) \(\approx\) \(1.04185 - 0.674091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.23 - 0.0131i)T \)
37 \( 1 + (1.57 - 5.87i)T \)
good2 \( 1 + (0.381 - 0.138i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (-1.04 + 2.85i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (-1.76 + 0.311i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (0.401 - 0.696i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.737 + 0.619i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (4.96 + 4.16i)T + (2.95 + 16.7i)T^{2} \)
19 \( 1 + (1.57 - 4.32i)T + (-14.5 - 12.2i)T^{2} \)
23 \( 1 + (-1.16 - 2.01i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.39 - 4.84i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.05iT - 31T^{2} \)
41 \( 1 + (-1.50 + 1.26i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + 8.69T + 43T^{2} \)
47 \( 1 + (-2.16 + 1.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.74 - 1.54i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-3.73 - 0.659i)T + (55.4 + 20.1i)T^{2} \)
61 \( 1 + (1.10 + 1.31i)T + (-10.5 + 60.0i)T^{2} \)
67 \( 1 + (12.3 - 2.17i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (12.4 + 4.52i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + 2.72iT - 73T^{2} \)
79 \( 1 + (-8.15 + 1.43i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.466 - 0.555i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-4.75 - 0.837i)T + (83.6 + 30.4i)T^{2} \)
97 \( 1 + (2.87 + 4.98i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.65295362039325854782806614737, −11.85121336240811530479836963157, −10.28994851353889115649984006596, −8.929408906017006675297895865481, −8.436146085172175722202993892058, −7.34966358700623336848548765982, −6.52155876048423582182080622177, −4.94234864571611990666227697949, −2.91394952271105387168588053333, −1.46515012859163654300371555801, 2.33548079448725510084952997883, 4.26209626328205523845242534571, 4.95637854344335492935082227758, 6.13068130023450235690068745341, 8.533532869213636891352763802039, 8.812732500287962301417106856899, 9.835862978973817561573373975447, 10.52637759279915991368139519591, 11.19582796704918853679503411385, 13.26627474640684129029005041825

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