Properties

Label 2-845-169.118-c1-0-1
Degree $2$
Conductor $845$
Sign $-0.832 + 0.554i$
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  + (−1.96 − 1.73i)2-s + (−0.904 + 2.38i)3-s + (0.585 + 4.82i)4-s + (0.970 + 0.239i)5-s + (5.91 − 3.10i)6-s + (−1.96 + 2.84i)7-s + (4.25 − 6.16i)8-s + (−2.62 − 2.32i)9-s + (−1.48 − 2.15i)10-s + (−4.47 + 3.96i)11-s + (−12.0 − 2.96i)12-s + (−0.554 − 3.56i)13-s + (8.80 − 2.17i)14-s + (−1.44 + 2.09i)15-s + (−9.61 + 2.37i)16-s + (−3.07 + 4.45i)17-s + ⋯
L(s)  = 1  + (−1.38 − 1.22i)2-s + (−0.522 + 1.37i)3-s + (0.292 + 2.41i)4-s + (0.434 + 0.107i)5-s + (2.41 − 1.26i)6-s + (−0.743 + 1.07i)7-s + (1.50 − 2.18i)8-s + (−0.875 − 0.775i)9-s + (−0.470 − 0.681i)10-s + (−1.35 + 1.19i)11-s + (−3.47 − 0.856i)12-s + (−0.153 − 0.988i)13-s + (2.35 − 0.580i)14-s + (−0.374 + 0.542i)15-s + (−2.40 + 0.592i)16-s + (−0.745 + 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.832 + 0.554i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (456, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.832 + 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0355240 - 0.117350i\)
\(L(\frac12)\) \(\approx\) \(0.0355240 - 0.117350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.970 - 0.239i)T \)
13 \( 1 + (0.554 + 3.56i)T \)
good2 \( 1 + (1.96 + 1.73i)T + (0.241 + 1.98i)T^{2} \)
3 \( 1 + (0.904 - 2.38i)T + (-2.24 - 1.98i)T^{2} \)
7 \( 1 + (1.96 - 2.84i)T + (-2.48 - 6.54i)T^{2} \)
11 \( 1 + (4.47 - 3.96i)T + (1.32 - 10.9i)T^{2} \)
17 \( 1 + (3.07 - 4.45i)T + (-6.02 - 15.8i)T^{2} \)
19 \( 1 - 0.227T + 19T^{2} \)
23 \( 1 + 6.41T + 23T^{2} \)
29 \( 1 + (-5.61 - 4.97i)T + (3.49 + 28.7i)T^{2} \)
31 \( 1 + (-8.77 + 4.60i)T + (17.6 - 25.5i)T^{2} \)
37 \( 1 + (-2.70 + 1.42i)T + (21.0 - 30.4i)T^{2} \)
41 \( 1 + (-1.29 + 3.41i)T + (-30.6 - 27.1i)T^{2} \)
43 \( 1 + (1.13 + 0.597i)T + (24.4 + 35.3i)T^{2} \)
47 \( 1 + (-0.189 + 1.55i)T + (-45.6 - 11.2i)T^{2} \)
53 \( 1 + (3.17 - 4.59i)T + (-18.7 - 49.5i)T^{2} \)
59 \( 1 + (-3.31 - 0.816i)T + (52.2 + 27.4i)T^{2} \)
61 \( 1 + (4.02 + 5.83i)T + (-21.6 + 57.0i)T^{2} \)
67 \( 1 + (-0.323 + 2.66i)T + (-65.0 - 16.0i)T^{2} \)
71 \( 1 + (3.26 - 8.61i)T + (-53.1 - 47.0i)T^{2} \)
73 \( 1 + (-10.8 + 9.58i)T + (8.79 - 72.4i)T^{2} \)
79 \( 1 + (1.02 - 8.44i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (-0.999 - 2.63i)T + (-62.1 + 55.0i)T^{2} \)
89 \( 1 - 5.43T + 89T^{2} \)
97 \( 1 + (8.56 - 2.11i)T + (85.8 - 45.0i)T^{2} \)
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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

−10.42319248774941971168865456379, −9.958366342281005445035475646222, −9.515835212901887282952956495723, −8.543282833457282549559887497073, −7.81349472011855083640740131052, −6.31837553018300510696598916254, −5.26250912345821133985994178758, −4.15443198692173082554870629502, −2.91074751276158934402152695952, −2.22357661577225583145661997943, 0.12136364809100842176150058130, 0.991422312664292559200423106901, 2.47573829708215794794585368860, 4.80927138091762059237239609282, 6.03874988651263237070756896263, 6.43271292190825229562033520451, 7.09605192965305919574286693080, 7.888740446520650140523379805087, 8.475727677270020966447923436651, 9.658505398223997608855955094502

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