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Properties

Label 2-185-185.103-c1-0-9
Degree $2$
Conductor $185$
Sign $0.810 - 0.585i$
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  + (1.07 + 1.85i)2-s + (−0.548 − 2.04i)3-s + (−1.30 + 2.26i)4-s + (2.02 − 0.948i)5-s + (3.22 − 3.22i)6-s + (0.242 + 0.905i)7-s − 1.31·8-s + (−1.29 + 0.749i)9-s + (3.93 + 2.74i)10-s − 2.13i·11-s + (5.35 + 1.43i)12-s + (−1.94 + 3.36i)13-s + (−1.42 + 1.42i)14-s + (−3.05 − 3.62i)15-s + (1.20 + 2.07i)16-s + (−1.44 + 0.832i)17-s + ⋯
L(s)  = 1  + (0.759 + 1.31i)2-s + (−0.316 − 1.18i)3-s + (−0.653 + 1.13i)4-s + (0.905 − 0.424i)5-s + (1.31 − 1.31i)6-s + (0.0916 + 0.342i)7-s − 0.464·8-s + (−0.432 + 0.249i)9-s + (1.24 + 0.869i)10-s − 0.645i·11-s + (1.54 + 0.413i)12-s + (−0.538 + 0.932i)13-s + (−0.380 + 0.380i)14-s + (−0.788 − 0.936i)15-s + (0.300 + 0.519i)16-s + (−0.349 + 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.810 - 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.810 - 0.585i$
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: \((2,\ 185,\ (\ :1/2),\ 0.810 - 0.585i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62712 + 0.525824i\)
\(L(\frac12)\) \(\approx\) \(1.62712 + 0.525824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.02 + 0.948i)T \)
37 \( 1 + (5.97 - 1.11i)T \)
good2 \( 1 + (-1.07 - 1.85i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.548 + 2.04i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.242 - 0.905i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + 2.13iT - 11T^{2} \)
13 \( 1 + (1.94 - 3.36i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.44 - 0.832i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.08 - 1.09i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.0404T + 23T^{2} \)
29 \( 1 + (0.157 - 0.157i)T - 29iT^{2} \)
31 \( 1 + (5.88 + 5.88i)T + 31iT^{2} \)
41 \( 1 + (-2.85 - 1.65i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.93T + 43T^{2} \)
47 \( 1 + (6.90 - 6.90i)T - 47iT^{2} \)
53 \( 1 + (3.52 - 13.1i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.72 + 6.43i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.53 + 0.410i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.80 - 0.484i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.50 + 12.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.92 - 1.92i)T - 73iT^{2} \)
79 \( 1 + (0.850 - 0.228i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (3.95 - 14.7i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-7.99 - 2.14i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−12.87708140787748651562198650240, −12.36121105765900899053749700503, −10.97353999671203439328113256377, −9.336381729951211105377495836133, −8.269771493167835092284435411955, −7.18272254799174642518945948869, −6.28153935199502893590039695295, −5.72057510710205754316018254730, −4.40995437097949172983155707328, −1.93440044487203459807403147157, 2.17326048279537296144691780012, 3.54792712375216884569600796547, 4.71080609641185579897941750547, 5.47341541930096128182764527981, 7.16112033840043508266880855166, 9.149363434176498834537024391213, 10.19182516318351783810465782909, 10.45493572946796625295896244994, 11.27307444709011109041018280909, 12.57361884817764362099273699370

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