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Properties

Label 2-845-65.9-c1-0-15
Degree $2$
Conductor $845$
Sign $0.991 + 0.126i$
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  + (−1.89 − 1.09i)2-s + (0.866 + 0.5i)3-s + (1.39 + 2.41i)4-s + (−2.18 + 0.456i)5-s + (−1.09 − 1.89i)6-s + (−1.5 + 0.866i)7-s − 1.73i·8-s + (−1 − 1.73i)9-s + (4.64 + 1.52i)10-s + (1.32 − 2.29i)11-s + 2.79i·12-s + 3.79·14-s + (−2.12 − 0.698i)15-s + (0.895 − 1.55i)16-s + (−3.96 + 2.29i)17-s + 4.37i·18-s + ⋯
L(s)  = 1  + (−1.34 − 0.773i)2-s + (0.499 + 0.288i)3-s + (0.697 + 1.20i)4-s + (−0.978 + 0.204i)5-s + (−0.446 − 0.773i)6-s + (−0.566 + 0.327i)7-s − 0.612i·8-s + (−0.333 − 0.577i)9-s + (1.47 + 0.483i)10-s + (0.398 − 0.690i)11-s + 0.805i·12-s + 1.01·14-s + (−0.548 − 0.180i)15-s + (0.223 − 0.387i)16-s + (−0.962 + 0.555i)17-s + 1.03i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591548 - 0.0375713i\)
\(L(\frac12)\) \(\approx\) \(0.591548 - 0.0375713i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.18 - 0.456i)T \)
13 \( 1 \)
good2 \( 1 + (1.89 + 1.09i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.866 - 0.5i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.5 - 0.866i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.32 + 2.29i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.96 - 2.29i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.866 + 1.5i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.96 - 2.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.29 - 3.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.66T + 31T^{2} \)
37 \( 1 + (-6.87 - 3.96i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.32 + 2.29i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.22 + 0.708i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.75iT - 47T^{2} \)
53 \( 1 - 1.58iT - 53T^{2} \)
59 \( 1 + (-1.68 - 2.91i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.29 - 9.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.8 - 7.43i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.77 + 3.08i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + (-2.14 + 3.70i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.87 - 2.23i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.04036206295382158367369641455, −9.196810101556079720504936594932, −8.718827493095998579177988704899, −8.124443805945757849282336785437, −7.02840511209263163279298839233, −6.09137780198639846324095373364, −4.37779141437114794348585996770, −3.27367463396918081345381706343, −2.68464434894639128991461990557, −0.842864669897239003186572351751, 0.63098984543899879391330488302, 2.33880702329700851019920244773, 3.79860969024694299915850814700, 4.91547309086091057347259521956, 6.49369757387865677275117814090, 6.99422699491478421161918363418, 7.87338404353843922915112040146, 8.332388718065606788836681427014, 9.179630951725924465980707162941, 9.846714628875843725896586845103

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