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Properties

Label 2-845-65.33-c1-0-42
Degree $2$
Conductor $845$
Sign $0.649 + 0.760i$
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  + (−0.759 + 1.31i)2-s + (−0.653 + 0.175i)3-s + (−0.152 − 0.263i)4-s + (−0.600 + 2.15i)5-s + (0.265 − 0.991i)6-s + (2.24 − 1.29i)7-s − 2.57·8-s + (−2.20 + 1.27i)9-s + (−2.37 − 2.42i)10-s + (−1.29 − 4.82i)11-s + (0.145 + 0.145i)12-s + 3.93i·14-s + (0.0150 − 1.51i)15-s + (2.25 − 3.91i)16-s + (0.0211 − 0.0790i)17-s − 3.85i·18-s + ⋯
L(s)  = 1  + (−0.536 + 0.929i)2-s + (−0.377 + 0.101i)3-s + (−0.0761 − 0.131i)4-s + (−0.268 + 0.963i)5-s + (0.108 − 0.404i)6-s + (0.849 − 0.490i)7-s − 0.910·8-s + (−0.733 + 0.423i)9-s + (−0.751 − 0.766i)10-s + (−0.390 − 1.45i)11-s + (0.0420 + 0.0420i)12-s + 1.05i·14-s + (0.00389 − 0.390i)15-s + (0.564 − 0.977i)16-s + (0.00513 − 0.0191i)17-s − 0.909i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.238167 - 0.109799i\)
\(L(\frac12)\) \(\approx\) \(0.238167 - 0.109799i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−10.23128843701834294857606890403, −8.825751557417481310870875919041, −8.169387701863593570283680475350, −7.69505898905472275628352858274, −6.59398505830797697184621863541, −6.02572595418672024707946249139, −5.01542253664367691397643020670, −3.58321787651689966106267188798, −2.55442356390229348401198791948, −0.15768520978635937358218438133, 1.44491545719096243363895554817, 2.32442177048564593445211173068, 3.85234629273812067005297877453, 5.09939652320686327381284499077, 5.64480190238016333313411333535, 6.94327404235107507219611567012, 8.158956564397710710435695953831, 8.729880505299240421603550608814, 9.574112849781042418993059247093, 10.26403538614610844380889081910

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