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Properties

Label 2-845-65.33-c1-0-47
Degree $2$
Conductor $845$
Sign $0.266 + 0.963i$
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  + (0.915 − 1.58i)2-s + (1.91 − 0.512i)3-s + (−0.677 − 1.17i)4-s + (−1.45 + 1.69i)5-s + (0.939 − 3.50i)6-s + (3.06 − 1.76i)7-s + 1.18·8-s + (0.803 − 0.463i)9-s + (1.36 + 3.86i)10-s + (1.00 + 3.74i)11-s + (−1.89 − 1.89i)12-s − 6.48i·14-s + (−1.91 + 3.99i)15-s + (2.43 − 4.22i)16-s + (0.524 − 1.95i)17-s − 1.69i·18-s + ⋯
L(s)  = 1  + (0.647 − 1.12i)2-s + (1.10 − 0.296i)3-s + (−0.338 − 0.586i)4-s + (−0.650 + 0.759i)5-s + (0.383 − 1.43i)6-s + (1.15 − 0.668i)7-s + 0.417·8-s + (0.267 − 0.154i)9-s + (0.430 + 1.22i)10-s + (0.302 + 1.12i)11-s + (−0.548 − 0.548i)12-s − 1.73i·14-s + (−0.494 + 1.03i)15-s + (0.609 − 1.05i)16-s + (0.127 − 0.474i)17-s − 0.400i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62511 - 1.99681i\)
\(L(\frac12)\) \(\approx\) \(2.62511 - 1.99681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.45 - 1.69i)T \)
13 \( 1 \)
good2 \( 1 + (-0.915 + 1.58i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.91 + 0.512i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-3.06 + 1.76i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.00 - 3.74i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.524 + 1.95i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.518 - 0.139i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.0788 - 0.294i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.71 - 0.988i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.13 + 4.13i)T + 31iT^{2} \)
37 \( 1 + (4.69 + 2.70i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.649 - 0.174i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (8.51 + 2.28i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 9.75iT - 47T^{2} \)
53 \( 1 + (-3.16 - 3.16i)T + 53iT^{2} \)
59 \( 1 + (-3.14 + 11.7i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.44 - 2.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.14 + 1.98i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.19 + 4.46i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 14.7T + 73T^{2} \)
79 \( 1 + 1.59iT - 79T^{2} \)
83 \( 1 - 7.57iT - 83T^{2} \)
89 \( 1 + (-4.54 + 1.21i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-8.91 - 15.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.29760016360969365138422886596, −9.335530540777469107845435996794, −8.100975353316205801811523987400, −7.58297563357195966197882355531, −6.96464185334971982266241877729, −5.06999605810188074431303717346, −4.19829214860436540843135031731, −3.46136288512181360571214763750, −2.45117700777851483328489642986, −1.60267395104275816436228449021, 1.64032157363093911691954204858, 3.31336784933127025960214804576, 4.17279983744360490316144516291, 5.13269475021819877710358263238, 5.76920243333591464881916221701, 7.03941288138552790235508061012, 8.060791113377232438456854063682, 8.503324610473645608422338843115, 8.857783240001120634396615417642, 10.26578347542730635842501488536

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