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Properties

Label 2-845-169.114-c1-0-55
Degree $2$
Conductor $845$
Sign $-0.780 + 0.624i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  + (0.370 + 0.302i)2-s + (0.900 + 0.0727i)3-s + (−0.354 − 1.73i)4-s + (−0.992 + 0.120i)5-s + (0.311 + 0.299i)6-s + (−1.50 − 2.38i)7-s + (0.838 − 1.59i)8-s + (−2.15 − 0.350i)9-s + (−0.404 − 0.255i)10-s + (−0.0262 − 0.161i)11-s + (−0.192 − 1.58i)12-s + (3.19 + 1.66i)13-s + (0.162 − 1.33i)14-s + (−0.902 + 0.0363i)15-s + (−2.46 + 1.05i)16-s + (−4.54 + 2.87i)17-s + ⋯
L(s)  = 1  + (0.261 + 0.213i)2-s + (0.520 + 0.0419i)3-s + (−0.177 − 0.867i)4-s + (−0.443 + 0.0539i)5-s + (0.127 + 0.122i)6-s + (−0.569 − 0.899i)7-s + (0.296 − 0.564i)8-s + (−0.718 − 0.116i)9-s + (−0.127 − 0.0808i)10-s + (−0.00790 − 0.0486i)11-s + (−0.0556 − 0.458i)12-s + (0.887 + 0.460i)13-s + (0.0434 − 0.357i)14-s + (−0.233 + 0.00939i)15-s + (−0.616 + 0.262i)16-s + (−1.10 + 0.696i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.780 + 0.624i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.780 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.310426 - 0.884815i\)
\(L(\frac12)\) \(\approx\) \(0.310426 - 0.884815i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.992 - 0.120i)T \)
13 \( 1 + (-3.19 - 1.66i)T \)
good2 \( 1 + (-0.370 - 0.302i)T + (0.400 + 1.95i)T^{2} \)
3 \( 1 + (-0.900 - 0.0727i)T + (2.96 + 0.481i)T^{2} \)
7 \( 1 + (1.50 + 2.38i)T + (-3.00 + 6.32i)T^{2} \)
11 \( 1 + (0.0262 + 0.161i)T + (-10.4 + 3.48i)T^{2} \)
17 \( 1 + (4.54 - 2.87i)T + (7.28 - 15.3i)T^{2} \)
19 \( 1 + (0.812 + 0.468i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.99 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.78 - 2.18i)T + (-5.80 - 28.4i)T^{2} \)
31 \( 1 + (-1.11 + 4.50i)T + (-27.4 - 14.4i)T^{2} \)
37 \( 1 + (10.0 - 2.89i)T + (31.2 - 19.7i)T^{2} \)
41 \( 1 + (-0.110 + 1.37i)T + (-40.4 - 6.57i)T^{2} \)
43 \( 1 + (-3.32 + 11.4i)T + (-36.3 - 22.9i)T^{2} \)
47 \( 1 + (-1.17 + 1.32i)T + (-5.66 - 46.6i)T^{2} \)
53 \( 1 + (-9.12 - 4.78i)T + (30.1 + 43.6i)T^{2} \)
59 \( 1 + (-2.97 + 6.97i)T + (-40.8 - 42.5i)T^{2} \)
61 \( 1 + (-0.291 + 7.23i)T + (-60.8 - 4.90i)T^{2} \)
67 \( 1 + (-10.8 - 2.22i)T + (61.6 + 26.2i)T^{2} \)
71 \( 1 + (9.23 - 4.38i)T + (44.9 - 54.9i)T^{2} \)
73 \( 1 + (-7.33 - 2.78i)T + (54.6 + 48.4i)T^{2} \)
79 \( 1 + (-6.82 - 6.04i)T + (9.52 + 78.4i)T^{2} \)
83 \( 1 + (11.6 - 8.07i)T + (29.4 - 77.6i)T^{2} \)
89 \( 1 + (-10.3 + 6.00i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.49 - 5.97i)T + (-26.9 - 93.1i)T^{2} \)
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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

−9.921687097197469354830204714484, −8.843652981490899508926132273791, −8.426856206025033135110160389511, −7.01617590838683900831464694194, −6.48705866076080002925953933922, −5.52253362522284449319816892549, −4.16655695354591373927436878912, −3.72357876724891711057382624458, −2.11335098054504931346541011360, −0.37049267238278419542356949909, 2.28603019399099807583184614966, 3.14146679312544280213918653735, 3.88688144714821622366752270495, 5.14223550715953710006833542253, 6.12084551458246393636930338860, 7.30282408512609202092975057940, 8.163887089318906254903550569940, 8.797572606200608784627343250969, 9.307663449882263834251620891812, 10.71192969629896451194987492420

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