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Properties

Label 2-185-185.103-c1-0-10
Degree $2$
Conductor $185$
Sign $0.961 - 0.273i$
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  + (0.0525 + 0.0909i)2-s + (0.518 + 1.93i)3-s + (0.994 − 1.72i)4-s + (2.22 + 0.266i)5-s + (−0.148 + 0.148i)6-s + (−1.16 − 4.36i)7-s + 0.419·8-s + (−0.881 + 0.508i)9-s + (0.0923 + 0.216i)10-s + 2.44i·11-s + (3.85 + 1.03i)12-s + (−2.32 + 4.02i)13-s + (0.335 − 0.335i)14-s + (0.635 + 4.43i)15-s + (−1.96 − 3.40i)16-s + (−6.61 + 3.81i)17-s + ⋯
L(s)  = 1  + (0.0371 + 0.0643i)2-s + (0.299 + 1.11i)3-s + (0.497 − 0.861i)4-s + (0.992 + 0.119i)5-s + (−0.0607 + 0.0607i)6-s + (−0.441 − 1.64i)7-s + 0.148·8-s + (−0.293 + 0.169i)9-s + (0.0292 + 0.0683i)10-s + 0.736i·11-s + (1.11 + 0.297i)12-s + (−0.644 + 1.11i)13-s + (0.0896 − 0.0896i)14-s + (0.164 + 1.14i)15-s + (−0.491 − 0.851i)16-s + (−1.60 + 0.926i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.961 - 0.273i$
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: \((2,\ 185,\ (\ :1/2),\ 0.961 - 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51659 + 0.211582i\)
\(L(\frac12)\) \(\approx\) \(1.51659 + 0.211582i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.22 - 0.266i)T \)
37 \( 1 + (5.61 - 2.33i)T \)
good2 \( 1 + (-0.0525 - 0.0909i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.518 - 1.93i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.16 + 4.36i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 + (2.32 - 4.02i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.61 - 3.81i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.19 + 0.321i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 0.150T + 23T^{2} \)
29 \( 1 + (0.503 - 0.503i)T - 29iT^{2} \)
31 \( 1 + (5.31 + 5.31i)T + 31iT^{2} \)
41 \( 1 + (-4.77 - 2.75i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 - 1.79T + 43T^{2} \)
47 \( 1 + (-1.65 + 1.65i)T - 47iT^{2} \)
53 \( 1 + (-3.03 + 11.3i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.0783 + 0.292i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.65 - 0.711i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-12.3 + 3.29i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-5.25 + 9.10i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.35 - 6.35i)T - 73iT^{2} \)
79 \( 1 + (-4.95 + 1.32i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (0.236 - 0.883i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-7.43 - 1.99i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 1.75iT - 97T^{2} \)
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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

−12.93663988630521554487906141042, −11.13897465806299191427016781117, −10.46378525331053817229865777031, −9.788565573332013971422235121021, −9.231550687016469608931377550556, −7.10537322831339272154940102329, −6.51684359806722303507665621237, −4.89817469170397324687707963385, −3.99657676279318173893091536547, −1.99034535216132790468633356207, 2.23716989245234744167278948969, 2.84289677277733811794897623633, 5.36288681442962826084648050046, 6.39634539889435006368792675338, 7.35261124783816268601150275737, 8.572042208589895103865203260135, 9.187644665223288301373522137096, 10.77425189634904791926562084051, 12.00452507890432584626870464343, 12.66277793828531162555122568129

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