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Properties

Label 2-1805-5.4-c1-0-11
Degree $2$
Conductor $1805$
Sign $-0.369 - 0.929i$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
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  + 1.93i·2-s − 2.41i·3-s − 1.74·4-s + (−0.825 − 2.07i)5-s + 4.67·6-s − 1.46i·7-s + 0.487i·8-s − 2.82·9-s + (4.02 − 1.59i)10-s − 2.89·11-s + 4.22i·12-s + 6.12i·13-s + 2.82·14-s + (−5.01 + 1.99i)15-s − 4.44·16-s + 6.29i·17-s + ⋯
L(s)  = 1  + 1.36i·2-s − 1.39i·3-s − 0.874·4-s + (−0.369 − 0.929i)5-s + 1.90·6-s − 0.552i·7-s + 0.172i·8-s − 0.942·9-s + (1.27 − 0.505i)10-s − 0.872·11-s + 1.21i·12-s + 1.69i·13-s + 0.755·14-s + (−1.29 + 0.514i)15-s − 1.11·16-s + 1.52i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.369 - 0.929i$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ -0.369 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9715202917\)
\(L(\frac12)\) \(\approx\) \(0.9715202917\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.825 + 2.07i)T \)
19 \( 1 \)
good2 \( 1 - 1.93iT - 2T^{2} \)
3 \( 1 + 2.41iT - 3T^{2} \)
7 \( 1 + 1.46iT - 7T^{2} \)
11 \( 1 + 2.89T + 11T^{2} \)
13 \( 1 - 6.12iT - 13T^{2} \)
17 \( 1 - 6.29iT - 17T^{2} \)
23 \( 1 - 0.508iT - 23T^{2} \)
29 \( 1 - 1.72T + 29T^{2} \)
31 \( 1 - 8.44T + 31T^{2} \)
37 \( 1 + 3.13iT - 37T^{2} \)
41 \( 1 + 7.44T + 41T^{2} \)
43 \( 1 - 6.90iT - 43T^{2} \)
47 \( 1 - 0.316iT - 47T^{2} \)
53 \( 1 - 4.34iT - 53T^{2} \)
59 \( 1 - 2.25T + 59T^{2} \)
61 \( 1 + 6.29T + 61T^{2} \)
67 \( 1 - 10.0iT - 67T^{2} \)
71 \( 1 - 6.63T + 71T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 + 4.94iT - 83T^{2} \)
89 \( 1 + 5.35T + 89T^{2} \)
97 \( 1 + 15.9iT - 97T^{2} \)
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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

−8.956205521305059871635668251294, −8.298002710810487318509962753891, −7.87579164123057459468674027313, −7.12705821134470258184727402476, −6.51658650737556213753410158433, −5.85672269618873061120829135113, −4.79782504411599625151421255313, −4.07635096785906442186349342451, −2.25411170900789102715648567711, −1.23807353618527167284167672091, 0.37952069487820784189390283966, 2.50722935078352745080656355863, 2.99983870154605288144258560799, 3.60674910445508336270635601187, 4.76947808051702145901672536359, 5.31219772805146889067295568438, 6.57190860215173121043170134584, 7.66495577981799939020959956693, 8.518929937442804629629941108971, 9.534139384041226059527743072612

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