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Properties

Label 2-845-65.37-c1-0-46
Degree $2$
Conductor $845$
Sign $0.550 + 0.834i$
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  + (1.31 − 0.759i)2-s + (−0.653 − 0.175i)3-s + (0.152 − 0.263i)4-s + (2.15 − 0.600i)5-s + (−0.991 + 0.265i)6-s + (1.29 − 2.24i)7-s + 2.57i·8-s + (−2.20 − 1.27i)9-s + (2.37 − 2.42i)10-s + (4.82 + 1.29i)11-s + (−0.145 + 0.145i)12-s − 3.93i·14-s + (−1.51 + 0.0150i)15-s + (2.25 + 3.91i)16-s + (−0.0211 − 0.0790i)17-s − 3.85·18-s + ⋯
L(s)  = 1  + (0.929 − 0.536i)2-s + (−0.377 − 0.101i)3-s + (0.0761 − 0.131i)4-s + (0.963 − 0.268i)5-s + (−0.404 + 0.108i)6-s + (0.490 − 0.849i)7-s + 0.910i·8-s + (−0.733 − 0.423i)9-s + (0.751 − 0.766i)10-s + (1.45 + 0.390i)11-s + (−0.0420 + 0.0420i)12-s − 1.05i·14-s + (−0.390 + 0.00389i)15-s + (0.564 + 0.977i)16-s + (−0.00513 − 0.0191i)17-s − 0.909·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 + 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.550 + 0.834i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.550 + 0.834i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35571 - 1.26845i\)
\(L(\frac12)\) \(\approx\) \(2.35571 - 1.26845i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.15 + 0.600i)T \)
13 \( 1 \)
good2 \( 1 + (-1.31 + 0.759i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.653 + 0.175i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.29 + 2.24i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.82 - 1.29i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.0211 + 0.0790i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.726 + 2.71i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.05 + 3.91i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (4.31 - 2.49i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.32 - 2.32i)T + 31iT^{2} \)
37 \( 1 + (-0.285 - 0.494i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.69 + 10.0i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.132 - 0.0354i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 2.30T + 47T^{2} \)
53 \( 1 + (-6.70 + 6.70i)T - 53iT^{2} \)
59 \( 1 + (2.59 - 0.694i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.74 - 4.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.6 - 7.89i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.42 - 1.98i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + 6.61iT - 73T^{2} \)
79 \( 1 - 5.71iT - 79T^{2} \)
83 \( 1 + 3.70T + 83T^{2} \)
89 \( 1 + (4.63 - 17.2i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-4.65 - 2.68i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.37689865774745240075309777691, −9.087425898700827798526934948957, −8.700609171972631808768935025903, −7.23441263780066662688576067453, −6.36563868551802132486744377028, −5.47324503241391808637106277448, −4.59552340167353895213159707003, −3.79714937141590995814973549097, −2.51438004034935702098593987162, −1.23736946958450352686670350882, 1.56914670491405406544045682754, 3.00604379210378082981054656107, 4.27195882641174546176685433048, 5.27221974282099532686599752958, 5.99667370497567152175019367901, 6.24314469280300030296274890466, 7.53491704604868062128816358625, 8.796502976910130941452832392531, 9.407000582145317195279407396981, 10.32388621521171108987550045741

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