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Properties

Label 2-845-65.47-c1-0-48
Degree $2$
Conductor $845$
Sign $0.594 - 0.804i$
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  + 2.08i·2-s + (1.94 − 1.94i)3-s − 2.35·4-s + (2.22 − 0.194i)5-s + (4.06 + 4.06i)6-s + 2.91·7-s − 0.750i·8-s − 4.59i·9-s + (0.405 + 4.65i)10-s + (0.0186 − 0.0186i)11-s + (−4.59 + 4.59i)12-s + 6.07i·14-s + (3.96 − 4.71i)15-s − 3.15·16-s + (−2.02 + 2.02i)17-s + 9.58·18-s + ⋯
L(s)  = 1  + 1.47i·2-s + (1.12 − 1.12i)3-s − 1.17·4-s + (0.996 − 0.0869i)5-s + (1.66 + 1.66i)6-s + 1.10·7-s − 0.265i·8-s − 1.53i·9-s + (0.128 + 1.47i)10-s + (0.00561 − 0.00561i)11-s + (−1.32 + 1.32i)12-s + 1.62i·14-s + (1.02 − 1.21i)15-s − 0.787·16-s + (−0.491 + 0.491i)17-s + 2.26·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.594 - 0.804i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.594 - 0.804i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.594 - 0.804i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.50083 + 1.26186i\)
\(L(\frac12)\) \(\approx\) \(2.50083 + 1.26186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.22 + 0.194i)T \)
13 \( 1 \)
good2 \( 1 - 2.08iT - 2T^{2} \)
3 \( 1 + (-1.94 + 1.94i)T - 3iT^{2} \)
7 \( 1 - 2.91T + 7T^{2} \)
11 \( 1 + (-0.0186 + 0.0186i)T - 11iT^{2} \)
17 \( 1 + (2.02 - 2.02i)T - 17iT^{2} \)
19 \( 1 + (3.38 - 3.38i)T - 19iT^{2} \)
23 \( 1 + (0.262 + 0.262i)T + 23iT^{2} \)
29 \( 1 + 4.18iT - 29T^{2} \)
31 \( 1 + (0.835 + 0.835i)T + 31iT^{2} \)
37 \( 1 - 6.45T + 37T^{2} \)
41 \( 1 + (5.54 + 5.54i)T + 41iT^{2} \)
43 \( 1 + (4.90 + 4.90i)T + 43iT^{2} \)
47 \( 1 - 0.833T + 47T^{2} \)
53 \( 1 + (-0.902 + 0.902i)T - 53iT^{2} \)
59 \( 1 + (-1.05 - 1.05i)T + 59iT^{2} \)
61 \( 1 + 10.7T + 61T^{2} \)
67 \( 1 - 12.3iT - 67T^{2} \)
71 \( 1 + (-2.61 - 2.61i)T + 71iT^{2} \)
73 \( 1 - 15.0iT - 73T^{2} \)
79 \( 1 + 4.25iT - 79T^{2} \)
83 \( 1 - 1.31T + 83T^{2} \)
89 \( 1 + (2.36 + 2.36i)T + 89iT^{2} \)
97 \( 1 + 0.405iT - 97T^{2} \)
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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

−9.951867017553171805244850807086, −8.813905847646557963981066150435, −8.447497864322084982861408337890, −7.78702231794377071995841617420, −6.95369824147396081395681343485, −6.22576551889017444404360075483, −5.39943406259429506968859101444, −4.22173236078510584434831895742, −2.41515821934855298408008568021, −1.65092760765377212907995331662, 1.63805660642004940165753746008, 2.50644810483556170965102575452, 3.29081574748365022366990285778, 4.54490193244923897502538709695, 4.91518743614196355549567876460, 6.57905524387494821394186814898, 8.011229594474805676618666514037, 8.940762567608604733632465713640, 9.343597246533216694689079553859, 10.11634447592171060746642296957

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