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Properties

Label 2-845-65.49-c1-0-36
Degree $2$
Conductor $845$
Sign $-0.998 - 0.0496i$
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  + (−1.09 − 1.89i)2-s + (−0.866 + 0.5i)3-s + (−1.39 + 2.41i)4-s + (0.456 + 2.18i)5-s + (1.89 + 1.09i)6-s + (−0.866 + 1.5i)7-s + 1.73·8-s + (−1 + 1.73i)9-s + (3.64 − 3.26i)10-s + (−2.29 + 1.32i)11-s − 2.79i·12-s + 3.79·14-s + (−1.49 − 1.66i)15-s + (0.895 + 1.55i)16-s + (−3.96 − 2.29i)17-s + 4.37·18-s + ⋯
L(s)  = 1  + (−0.773 − 1.34i)2-s + (−0.499 + 0.288i)3-s + (−0.697 + 1.20i)4-s + (0.204 + 0.978i)5-s + (0.773 + 0.446i)6-s + (−0.327 + 0.566i)7-s + 0.612·8-s + (−0.333 + 0.577i)9-s + (1.15 − 1.03i)10-s + (−0.690 + 0.398i)11-s − 0.805i·12-s + 1.01·14-s + (−0.384 − 0.430i)15-s + (0.223 + 0.387i)16-s + (−0.962 − 0.555i)17-s + 1.03·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.998 - 0.0496i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.998 - 0.0496i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00215265 + 0.0867318i\)
\(L(\frac12)\) \(\approx\) \(0.00215265 + 0.0867318i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−9.875592214008461843319584686809, −9.392263327060597588742237250506, −8.326598649507811020571420994178, −7.38035391925266846683928008359, −6.20596628392629979070321082110, −5.30835426588390338535362101913, −3.97424646245423033787612340324, −2.61898741687283792677495072588, −2.29987392487499704433964898739, −0.06426892555019445991838729672, 1.16315724230601871408103542684, 3.35416812609135171947488499030, 4.96584079784842853529241997381, 5.52840432378705674077157988003, 6.56031580011922173441556033428, 7.01996520328485253524527797422, 8.167231499036962428791065935333, 8.755788616464658674412770104950, 9.416715128572746720730187061596, 10.36363840949600263953071865251

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