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Properties

Label 2-845-5.4-c1-0-46
Degree $2$
Conductor $845$
Sign $0.949 - 0.314i$
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  − 0.330i·2-s + 2.69i·3-s + 1.89·4-s + (2.12 − 0.702i)5-s + 0.890·6-s − 3.35i·7-s − 1.28i·8-s − 4.24·9-s + (−0.232 − 0.702i)10-s + 3.24·11-s + 5.08i·12-s − 1.10·14-s + (1.89 + 5.71i)15-s + 3.35·16-s − 1.94i·17-s + 1.40i·18-s + ⋯
L(s)  = 1  − 0.233i·2-s + 1.55i·3-s + 0.945·4-s + (0.949 − 0.314i)5-s + 0.363·6-s − 1.26i·7-s − 0.455i·8-s − 1.41·9-s + (−0.0734 − 0.222i)10-s + 0.978·11-s + 1.46i·12-s − 0.296·14-s + (0.488 + 1.47i)15-s + 0.838·16-s − 0.472i·17-s + 0.331i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.949 - 0.314i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.949 - 0.314i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33152 + 0.375664i\)
\(L(\frac12)\) \(\approx\) \(2.33152 + 0.375664i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−10.39455265423008121207997558000, −9.530351234546954549848254149465, −9.039955838467582659863339747438, −7.63108227310904749343211440999, −6.63687805923797099231566977272, −5.82448093135853213133970765551, −4.64812175091964646249489584289, −3.93206633659778725023937943193, −2.87697340493453317000613467535, −1.35841186430566095559491019844, 1.70593567998777809041882747998, 2.05938112293873316572744748749, 3.23125180571756484912924681301, 5.49052931393807979022730875298, 5.99134887369645766735046624202, 6.70060150470449406831225465827, 7.27613493906560028839142204185, 8.378156002596949713306408880061, 9.053377001628968638794316522912, 10.17815860723487155849629862538

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