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Properties

Label 2-845-65.18-c1-0-38
Degree $2$
Conductor $845$
Sign $0.889 + 0.457i$
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  + 0.131i·2-s + (−0.243 − 0.243i)3-s + 1.98·4-s + (2.08 − 0.813i)5-s + (0.0319 − 0.0319i)6-s − 2.78·7-s + 0.522i·8-s − 2.88i·9-s + (0.106 + 0.273i)10-s + (2.86 + 2.86i)11-s + (−0.482 − 0.482i)12-s − 0.365i·14-s + (−0.704 − 0.308i)15-s + 3.89·16-s + (−1.71 − 1.71i)17-s + 0.378·18-s + ⋯
L(s)  = 1  + 0.0928i·2-s + (−0.140 − 0.140i)3-s + 0.991·4-s + (0.931 − 0.363i)5-s + (0.0130 − 0.0130i)6-s − 1.05·7-s + 0.184i·8-s − 0.960i·9-s + (0.0337 + 0.0864i)10-s + (0.864 + 0.864i)11-s + (−0.139 − 0.139i)12-s − 0.0976i·14-s + (−0.181 − 0.0797i)15-s + 0.974·16-s + (−0.415 − 0.415i)17-s + 0.0891·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.889 + 0.457i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (408, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.889 + 0.457i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03433 - 0.492860i\)
\(L(\frac12)\) \(\approx\) \(2.03433 - 0.492860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.08 + 0.813i)T \)
13 \( 1 \)
good2 \( 1 - 0.131iT - 2T^{2} \)
3 \( 1 + (0.243 + 0.243i)T + 3iT^{2} \)
7 \( 1 + 2.78T + 7T^{2} \)
11 \( 1 + (-2.86 - 2.86i)T + 11iT^{2} \)
17 \( 1 + (1.71 + 1.71i)T + 17iT^{2} \)
19 \( 1 + (-1.34 - 1.34i)T + 19iT^{2} \)
23 \( 1 + (-5.64 + 5.64i)T - 23iT^{2} \)
29 \( 1 - 4.57iT - 29T^{2} \)
31 \( 1 + (-3.87 + 3.87i)T - 31iT^{2} \)
37 \( 1 + 7.01T + 37T^{2} \)
41 \( 1 + (4.54 - 4.54i)T - 41iT^{2} \)
43 \( 1 + (-4.57 + 4.57i)T - 43iT^{2} \)
47 \( 1 + 0.512T + 47T^{2} \)
53 \( 1 + (1.32 + 1.32i)T + 53iT^{2} \)
59 \( 1 + (1.85 - 1.85i)T - 59iT^{2} \)
61 \( 1 + 1.28T + 61T^{2} \)
67 \( 1 + 3.61iT - 67T^{2} \)
71 \( 1 + (-4.54 + 4.54i)T - 71iT^{2} \)
73 \( 1 - 9.93iT - 73T^{2} \)
79 \( 1 - 8.37iT - 79T^{2} \)
83 \( 1 - 3.17T + 83T^{2} \)
89 \( 1 + (-4.40 + 4.40i)T - 89iT^{2} \)
97 \( 1 - 11.7iT - 97T^{2} \)
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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

−9.981540371325120053222375645422, −9.422788302659769467106530812562, −8.648092636030954237378339459108, −7.08635580396525122091410289180, −6.66002858123836291626779113626, −6.09831572687956600813385778047, −4.93327233563966699460240420652, −3.51201364127927109765984051833, −2.46395682183697622778210120539, −1.17777456569643888648088585316, 1.52412774748482439328381123465, 2.74193494047716484393546360332, 3.51651701643033565557865486144, 5.18500054467159828639152312038, 6.08599032058861460430693505672, 6.63467205640087584197647479401, 7.45629296309505698644327028401, 8.736717114145014693816185343487, 9.583356007920477538270835127122, 10.35481248073282101191468064040

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