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Properties

Label 2-845-13.12-c1-0-38
Degree $2$
Conductor $845$
Sign $-0.554 + 0.832i$
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  − 1.30i·2-s + 3-s + 0.302·4-s + i·5-s − 1.30i·6-s − i·7-s − 3i·8-s − 2·9-s + 1.30·10-s − 5.60i·11-s + 0.302·12-s − 1.30·14-s + i·15-s − 3.30·16-s − 0.394·17-s + 2.60i·18-s + ⋯
L(s)  = 1  − 0.921i·2-s + 0.577·3-s + 0.151·4-s + 0.447i·5-s − 0.531i·6-s − 0.377i·7-s − 1.06i·8-s − 0.666·9-s + 0.411·10-s − 1.69i·11-s + 0.0874·12-s − 0.348·14-s + 0.258i·15-s − 0.825·16-s − 0.0956·17-s + 0.614i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900838 - 1.68323i\)
\(L(\frac12)\) \(\approx\) \(0.900838 - 1.68323i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 + 1.30iT - 2T^{2} \)
3 \( 1 - T + 3T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 + 5.60iT - 11T^{2} \)
17 \( 1 + 0.394T + 17T^{2} \)
19 \( 1 + 1.60iT - 19T^{2} \)
23 \( 1 - 3T + 23T^{2} \)
29 \( 1 - 8.21T + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + 3.60iT - 37T^{2} \)
41 \( 1 + 3iT - 41T^{2} \)
43 \( 1 + 4.21T + 43T^{2} \)
47 \( 1 + 5.21iT - 47T^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 - 10.8iT - 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 7iT - 67T^{2} \)
71 \( 1 - 16.8iT - 71T^{2} \)
73 \( 1 + 15.2iT - 73T^{2} \)
79 \( 1 + 9.21T + 79T^{2} \)
83 \( 1 + 5.21iT - 83T^{2} \)
89 \( 1 - 8.21iT - 89T^{2} \)
97 \( 1 - 15.6iT - 97T^{2} \)
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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

−10.27054069105437725934949483021, −8.980216828612454222552741662373, −8.522671620932501222916620042011, −7.33783845192385672678610007151, −6.50498729328812230102940576659, −5.52035299801190722034314116864, −3.93929568582082845433700094046, −3.12614132770711678065679383745, −2.51561191865469427255836164011, −0.859888855228293190432234916455, 1.93123094981391850820771288660, 2.87673393773674537621593673624, 4.46615198507554606612259603737, 5.27961296603731997183166957700, 6.25447385130200532533504173511, 7.11750543823000065923678209129, 7.949827979854183553008647373506, 8.569863572290455158754656281074, 9.390240678583582840597506413423, 10.26849664446102584539420330658

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