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Properties

Label 2-1805-95.14-c0-0-0
Degree $2$
Conductor $1805$
Sign $-0.845 + 0.533i$
Analytic cond. $0.900812$
Root an. cond. $0.949111$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 + 0.483i)2-s + (−0.245 + 1.39i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.347 − 1.96i)6-s + (−0.939 − 0.342i)9-s + (1.32 + 0.483i)10-s + (0.707 + 1.22i)12-s + (0.245 + 1.39i)13-s + (1.08 − 0.909i)15-s + (−0.173 + 0.984i)16-s + 1.41·18-s − 0.999·20-s + (0.173 + 0.984i)25-s + (−1 − 1.73i)26-s + ⋯
L(s)  = 1  + (−1.32 + 0.483i)2-s + (−0.245 + 1.39i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.347 − 1.96i)6-s + (−0.939 − 0.342i)9-s + (1.32 + 0.483i)10-s + (0.707 + 1.22i)12-s + (0.245 + 1.39i)13-s + (1.08 − 0.909i)15-s + (−0.173 + 0.984i)16-s + 1.41·18-s − 0.999·20-s + (0.173 + 0.984i)25-s + (−1 − 1.73i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.845 + 0.533i$
Analytic conductor: \(0.900812\)
Root analytic conductor: \(0.949111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :0),\ -0.845 + 0.533i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2401954286\)
\(L(\frac12)\) \(\approx\) \(0.2401954286\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 \)
good2 \( 1 + (1.32 - 0.483i)T + (0.766 - 0.642i)T^{2} \)
3 \( 1 + (0.245 - 1.39i)T + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.245 - 1.39i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + 1.41T + T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (1.08 - 0.909i)T + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (1.32 + 0.483i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (-1.32 + 0.483i)T + (0.766 - 0.642i)T^{2} \)
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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

−9.628113830277315792486092554161, −9.104546899507887812507173730861, −8.683175503853072058820068548487, −7.77508663169256545467467060275, −7.00577824618087238077983433393, −6.04462152905569660473047649571, −4.86317289023032285070880744562, −4.28362675037257687350990296068, −3.48016020576548045932467901397, −1.53668442124365781537246976097, 0.30988814033562018342300677053, 1.49542980822512303149410833942, 2.57222163093689600475638201888, 3.48123399173255584677050000874, 5.07742555959086359919096700845, 6.16524261718607018767844791328, 7.01696936529625044106680819630, 7.59609151737841129863245964996, 8.153729640480690776247024037008, 8.725803663276601776073560168225

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