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Properties

Label 2-805-7.2-c1-0-18
Degree $2$
Conductor $805$
Sign $0.0230 - 0.999i$
Analytic cond. $6.42795$
Root an. cond. $2.53534$
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  + (0.270 − 0.468i)2-s + (1.08 + 1.87i)3-s + (0.853 + 1.47i)4-s + (0.5 − 0.866i)5-s + 1.17·6-s + (−1.51 + 2.16i)7-s + 2.00·8-s + (−0.844 + 1.46i)9-s + (−0.270 − 0.468i)10-s + (−0.0442 − 0.0765i)11-s + (−1.84 + 3.20i)12-s − 2.83·13-s + (0.605 + 1.29i)14-s + 2.16·15-s + (−1.16 + 2.01i)16-s + (3.34 + 5.78i)17-s + ⋯
L(s)  = 1  + (0.191 − 0.331i)2-s + (0.625 + 1.08i)3-s + (0.426 + 0.739i)4-s + (0.223 − 0.387i)5-s + 0.477·6-s + (−0.572 + 0.819i)7-s + 0.708·8-s + (−0.281 + 0.487i)9-s + (−0.0854 − 0.148i)10-s + (−0.0133 − 0.0230i)11-s + (−0.533 + 0.924i)12-s − 0.787·13-s + (0.161 + 0.346i)14-s + 0.559·15-s + (−0.291 + 0.504i)16-s + (0.810 + 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0230 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0230 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $0.0230 - 0.999i$
Analytic conductor: \(6.42795\)
Root analytic conductor: \(2.53534\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (576, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 805,\ (\ :1/2),\ 0.0230 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62404 + 1.58703i\)
\(L(\frac12)\) \(\approx\) \(1.62404 + 1.58703i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.51 - 2.16i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.270 + 0.468i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.08 - 1.87i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.0442 + 0.0765i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.83T + 13T^{2} \)
17 \( 1 + (-3.34 - 5.78i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.11 + 3.66i)T + (-9.5 - 16.4i)T^{2} \)
29 \( 1 + 4.39T + 29T^{2} \)
31 \( 1 + (0.375 + 0.650i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.43 + 2.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.21T + 41T^{2} \)
43 \( 1 - 2.09T + 43T^{2} \)
47 \( 1 + (-1.06 + 1.84i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.03 + 8.71i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.94 - 6.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.88 + 5.00i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.06 - 5.31i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (-3.93 - 6.81i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.05 + 13.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.61T + 83T^{2} \)
89 \( 1 + (-1.60 + 2.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.75T + 97T^{2} \)
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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

−10.27943710659803259734371089118, −9.645954954235152624290779389515, −8.896930952302964300952280907462, −8.160169784202790949578868721206, −7.13208570628195070664253546668, −5.91333343491379953967474181538, −4.88137569117098330435210503243, −3.83207628349377592926355364666, −3.12893899513218741596294662966, −2.10518265069707572930420465935, 1.02143372479139923177050371413, 2.25093774107949100045835937879, 3.26360303626128223684015223736, 4.81952142258658819456533499218, 5.88331557091397035651834864939, 6.78964509450042994840370269195, 7.39422449684728561964652287641, 7.81103339330269528110824038696, 9.407271919764770455352638370576, 9.954478909857605156452434758656

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