Properties

Label 2-845-65.64-c1-0-67
Degree $2$
Conductor $845$
Sign $-0.265 - 0.964i$
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  + 0.330·2-s − 2.69i·3-s − 1.89·4-s + (0.702 − 2.12i)5-s − 0.890i·6-s − 3.35·7-s − 1.28·8-s − 4.24·9-s + (0.232 − 0.702i)10-s + 3.24i·11-s + 5.08i·12-s − 1.10·14-s + (−5.71 − 1.89i)15-s + 3.35·16-s − 1.94i·17-s − 1.40·18-s + ⋯
L(s)  = 1  + 0.233·2-s − 1.55i·3-s − 0.945·4-s + (0.314 − 0.949i)5-s − 0.363i·6-s − 1.26·7-s − 0.455·8-s − 1.41·9-s + (0.0734 − 0.222i)10-s + 0.978i·11-s + 1.46i·12-s − 0.296·14-s + (−1.47 − 0.488i)15-s + 0.838·16-s − 0.472i·17-s − 0.331·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.265 - 0.964i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (844, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.177619 + 0.233091i\)
\(L(\frac12)\) \(\approx\) \(0.177619 + 0.233091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.702 + 2.12i)T \)
13 \( 1 \)
good2 \( 1 - 0.330T + 2T^{2} \)
3 \( 1 + 2.69iT - 3T^{2} \)
7 \( 1 + 3.35T + 7T^{2} \)
11 \( 1 - 3.24iT - 11T^{2} \)
17 \( 1 + 1.94iT - 17T^{2} \)
19 \( 1 - 1.24iT - 19T^{2} \)
23 \( 1 + 2.69iT - 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 3.78iT - 31T^{2} \)
37 \( 1 + 1.94T + 37T^{2} \)
41 \( 1 - 2.78iT - 41T^{2} \)
43 \( 1 - 8.73iT - 43T^{2} \)
47 \( 1 - 6.86T + 47T^{2} \)
53 \( 1 + 12.8iT - 53T^{2} \)
59 \( 1 + 2.53iT - 59T^{2} \)
61 \( 1 + 7.49T + 61T^{2} \)
67 \( 1 + 4.01T + 67T^{2} \)
71 \( 1 - 5.24iT - 71T^{2} \)
73 \( 1 + 5.46T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 8.61T + 83T^{2} \)
89 \( 1 - 10.3iT - 89T^{2} \)
97 \( 1 + 5.26T + 97T^{2} \)
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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

−9.497138810674034848490473514436, −8.725555685625146356730882562598, −7.890329284751843537466754430191, −6.93760545140345413108401804911, −6.15232854907396861210295065585, −5.26608865384886367294165568379, −4.20710315843859247182333210864, −2.83295926144566163802977199215, −1.45369984513997998457545750194, −0.13822346663812735828696757506, 2.96630423732411377861781913962, 3.55486020044067115627343432485, 4.28302045383937355074003488973, 5.66863886070168639784665007417, 5.95213368968777613633446564407, 7.33498282703170919948240533139, 8.765663551293161981340396306536, 9.230215635948525803147379466558, 9.997083776927163471638229936179, 10.48597983136445983333838486173

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