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Properties

Label 2-845-65.57-c1-0-31
Degree $2$
Conductor $845$
Sign $0.979 + 0.202i$
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-s + (1 − i)3-s − 4-s + (1 − 2i)5-s + (−1 + i)6-s + 2i·7-s + 3·8-s + i·9-s + (−1 + 2i)10-s + (1 + i)11-s + (−1 + i)12-s − 2i·14-s + (−1 − 3i)15-s − 16-s + (1 − i)17-s − i·18-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.577 − 0.577i)3-s − 0.5·4-s + (0.447 − 0.894i)5-s + (−0.408 + 0.408i)6-s + 0.755i·7-s + 1.06·8-s + 0.333i·9-s + (−0.316 + 0.632i)10-s + (0.301 + 0.301i)11-s + (−0.288 + 0.288i)12-s − 0.534i·14-s + (−0.258 − 0.774i)15-s − 0.250·16-s + (0.242 − 0.242i)17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.979 + 0.202i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.979 + 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28332 - 0.131403i\)
\(L(\frac12)\) \(\approx\) \(1.28332 - 0.131403i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1 + 2i)T \)
13 \( 1 \)
good2 \( 1 + T + 2T^{2} \)
3 \( 1 + (-1 + i)T - 3iT^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + (-1 - i)T + 11iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 + (-5 - 5i)T + 19iT^{2} \)
23 \( 1 + (-3 - 3i)T + 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (5 - 5i)T - 31iT^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + (-7 + 7i)T - 41iT^{2} \)
43 \( 1 + (1 + i)T + 43iT^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (-7 + 7i)T - 59iT^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (1 - i)T - 71iT^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (5 - 5i)T - 89iT^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.771603836877251884773775302253, −9.200243757568709178948787298780, −8.589892060628199988297932936725, −7.86393576383177772408631645189, −7.12013911552666554509174432922, −5.53830459405669053066837003804, −5.08139319533093154557660687378, −3.67179757851147163288609765016, −2.10953878103430875221379245190, −1.21084142230392679343381123098, 0.972144737925849040283467433381, 2.78812498195416515094628135212, 3.74485120437661299109604353213, 4.63319390685027673512157737634, 5.95641936729707310798056473999, 7.07648791172497236625123409406, 7.70508098683079452437475299650, 8.869807578067059749431236062820, 9.391333516584839524505667349951, 9.997695735296556593516238364177

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