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Properties

Label 2-845-13.4-c1-0-35
Degree $2$
Conductor $845$
Sign $-0.890 + 0.454i$
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  + (−2.09 − 1.20i)2-s + (0.707 − 1.22i)3-s + (1.91 + 3.31i)4-s − i·5-s + (−2.95 + 1.70i)6-s + (4.18 − 2.41i)7-s − 4.41i·8-s + (0.500 + 0.866i)9-s + (−1.20 + 2.09i)10-s + (−2.95 − 1.70i)11-s + 5.41·12-s − 11.6·14-s + (−1.22 − 0.707i)15-s + (−1.49 + 2.59i)16-s + (0.414 + 0.717i)17-s − 2.41i·18-s + ⋯
L(s)  = 1  + (−1.47 − 0.853i)2-s + (0.408 − 0.707i)3-s + (0.957 + 1.65i)4-s − 0.447i·5-s + (−1.20 + 0.696i)6-s + (1.58 − 0.912i)7-s − 1.56i·8-s + (0.166 + 0.288i)9-s + (−0.381 + 0.661i)10-s + (−0.891 − 0.514i)11-s + 1.56·12-s − 3.11·14-s + (−0.316 − 0.182i)15-s + (−0.374 + 0.649i)16-s + (0.100 + 0.174i)17-s − 0.569i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.890 + 0.454i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.890 + 0.454i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.225402 - 0.937260i\)
\(L(\frac12)\) \(\approx\) \(0.225402 - 0.937260i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.09 + 1.20i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.707 + 1.22i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-4.18 + 2.41i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.95 + 1.70i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.414 - 0.717i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.507 - 0.292i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.707 + 1.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.82 + 4.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.75iT - 31T^{2} \)
37 \( 1 + (-7.34 - 4.24i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.74 + 1.58i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.53 + 9.58i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.82iT - 47T^{2} \)
53 \( 1 - 2.48T + 53T^{2} \)
59 \( 1 + (-1.52 + 0.878i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.73 + i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.3 - 5.94i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.48iT - 73T^{2} \)
79 \( 1 + 8.48T + 79T^{2} \)
83 \( 1 - 3.17iT - 83T^{2} \)
89 \( 1 + (5.19 + 3i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.63 + 3.82i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.12388366341781794506072807564, −8.650378343610156615281229107311, −8.282110461023274592350792168370, −7.74249824961464884603840674970, −7.03981577214759012540601212813, −5.30463358560566370648080221339, −4.19515370526665064897974665810, −2.62548682188398569984388159428, −1.72240361859482729308624123089, −0.796103750884063539129003391681, 1.54099289816793069388234036081, 2.80964011967509439720999739821, 4.55859540323865933357535143955, 5.39545372223772320065199846992, 6.48702111410938815006721105134, 7.52461194068310709884317713748, 8.092270062746130344556634946057, 8.835020636018178689944449090427, 9.487718674683882005381340680295, 10.29307752681707937760663954906

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