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Properties

Label 2-845-65.32-c1-0-47
Degree $2$
Conductor $845$
Sign $-0.151 + 0.988i$
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  + (1.32 − 2.29i)2-s + (0.335 + 1.25i)3-s + (−2.51 − 4.34i)4-s + (1.30 + 1.81i)5-s + (3.31 + 0.889i)6-s + (0.0972 − 0.0561i)7-s − 8.00·8-s + (1.14 − 0.658i)9-s + (5.89 − 0.585i)10-s + (1.78 − 0.479i)11-s + (4.60 − 4.60i)12-s − 0.297i·14-s + (−1.83 + 2.24i)15-s + (−5.58 + 9.67i)16-s + (2.63 + 0.706i)17-s − 3.49i·18-s + ⋯
L(s)  = 1  + (0.936 − 1.62i)2-s + (0.193 + 0.723i)3-s + (−1.25 − 2.17i)4-s + (0.583 + 0.812i)5-s + (1.35 + 0.363i)6-s + (0.0367 − 0.0212i)7-s − 2.83·8-s + (0.380 − 0.219i)9-s + (1.86 − 0.185i)10-s + (0.539 − 0.144i)11-s + (1.32 − 1.32i)12-s − 0.0795i·14-s + (−0.474 + 0.579i)15-s + (−1.39 + 2.41i)16-s + (0.639 + 0.171i)17-s − 0.823i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.151 + 0.988i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.151 + 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89033 - 2.20246i\)
\(L(\frac12)\) \(\approx\) \(1.89033 - 2.20246i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.30 - 1.81i)T \)
13 \( 1 \)
good2 \( 1 + (-1.32 + 2.29i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.335 - 1.25i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (-0.0972 + 0.0561i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.78 + 0.479i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.63 - 0.706i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.80 + 6.72i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.10 + 0.831i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (4.03 + 2.32i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.624 + 0.624i)T - 31iT^{2} \)
37 \( 1 + (-1.27 - 0.737i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.40 + 5.24i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.00 - 3.76i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 0.345iT - 47T^{2} \)
53 \( 1 + (3.59 - 3.59i)T - 53iT^{2} \)
59 \( 1 + (1.24 + 0.332i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.39 - 2.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.0721 - 0.124i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.28 + 1.41i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 9.06T + 73T^{2} \)
79 \( 1 - 15.1iT - 79T^{2} \)
83 \( 1 - 8.53iT - 83T^{2} \)
89 \( 1 + (0.147 + 0.549i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (7.48 + 12.9i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.08131896223420051157823999506, −9.569718718599766744194494391138, −8.962592964488677947014836921421, −7.15292242865437169307860493769, −6.10797936985328110828503940873, −5.13571069131743323777397297444, −4.25794536227236698985234775791, −3.36141033169689710502551267897, −2.65152251112209191334173104521, −1.29153401544000360916509654711, 1.58588763845296692621616874732, 3.39867823083885514226506457164, 4.49131992088152941422634189942, 5.32947686957878927261186393757, 6.06191947662110188752811191671, 6.92213396767192548054717497296, 7.69708504806201772758498255182, 8.308098274931752324338297622556, 9.199366906562872408760308068298, 10.09539361837613217553247393789

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