Properties

Label 2-845-65.18-c1-0-4
Degree $2$
Conductor $845$
Sign $-0.999 + 0.0132i$
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  + 1.02i·2-s + (−1.97 − 1.97i)3-s + 0.954·4-s + (−1.45 + 1.69i)5-s + (2.01 − 2.01i)6-s + 0.963·7-s + 3.02i·8-s + 4.76i·9-s + (−1.73 − 1.49i)10-s + (−1.17 − 1.17i)11-s + (−1.88 − 1.88i)12-s + 0.985i·14-s + (6.21 − 0.469i)15-s − 1.18·16-s + (−5.12 − 5.12i)17-s − 4.87·18-s + ⋯
L(s)  = 1  + 0.723i·2-s + (−1.13 − 1.13i)3-s + 0.477·4-s + (−0.651 + 0.758i)5-s + (0.822 − 0.822i)6-s + 0.364·7-s + 1.06i·8-s + 1.58i·9-s + (−0.548 − 0.471i)10-s + (−0.354 − 0.354i)11-s + (−0.542 − 0.542i)12-s + 0.263i·14-s + (1.60 − 0.121i)15-s − 0.295·16-s + (−1.24 − 1.24i)17-s − 1.14·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.999 + 0.0132i$
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: \((2,\ 845,\ (\ :1/2),\ -0.999 + 0.0132i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00141585 - 0.213150i\)
\(L(\frac12)\) \(\approx\) \(0.00141585 - 0.213150i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.45 - 1.69i)T \)
13 \( 1 \)
good2 \( 1 - 1.02iT - 2T^{2} \)
3 \( 1 + (1.97 + 1.97i)T + 3iT^{2} \)
7 \( 1 - 0.963T + 7T^{2} \)
11 \( 1 + (1.17 + 1.17i)T + 11iT^{2} \)
17 \( 1 + (5.12 + 5.12i)T + 17iT^{2} \)
19 \( 1 + (-1.93 - 1.93i)T + 19iT^{2} \)
23 \( 1 + (2.72 - 2.72i)T - 23iT^{2} \)
29 \( 1 - 0.292iT - 29T^{2} \)
31 \( 1 + (0.125 - 0.125i)T - 31iT^{2} \)
37 \( 1 + 4.08T + 37T^{2} \)
41 \( 1 + (4.89 - 4.89i)T - 41iT^{2} \)
43 \( 1 + (5.62 - 5.62i)T - 43iT^{2} \)
47 \( 1 + 7.84T + 47T^{2} \)
53 \( 1 + (1.99 + 1.99i)T + 53iT^{2} \)
59 \( 1 + (-3.57 + 3.57i)T - 59iT^{2} \)
61 \( 1 - 2.08T + 61T^{2} \)
67 \( 1 + 7.29iT - 67T^{2} \)
71 \( 1 + (9.22 - 9.22i)T - 71iT^{2} \)
73 \( 1 - 3.22iT - 73T^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 + 8.56T + 83T^{2} \)
89 \( 1 + (0.366 - 0.366i)T - 89iT^{2} \)
97 \( 1 + 7.51iT - 97T^{2} \)
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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

−11.14056455368171440607474333935, −9.990905476249663546863022336010, −8.336274295441259602038865971145, −7.77764772191212822831158234392, −6.93714119647226015065404439703, −6.61270012216570025631573975472, −5.65008519634064258343521606477, −4.81863203137735381048560388313, −3.02907651278299000243045591810, −1.77805904415179210057863097187, 0.11245504545881684419442684407, 1.81451616194069955192002821659, 3.52149404740169573274284977525, 4.35192127243358550343548620860, 5.02273412663192977759016574189, 6.09577923643428774782518839873, 7.04985750531035555246778886246, 8.276305452750931474902882156674, 9.179214110885652967231380758764, 10.23994849463994907843421649180

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