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Properties

Label 2-845-65.4-c1-0-65
Degree $2$
Conductor $845$
Sign $-0.992 - 0.118i$
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  + (1.27 − 2.20i)2-s + (1.86 + 1.07i)3-s + (−2.24 − 3.88i)4-s + (−2.08 − 0.817i)5-s + (4.74 − 2.74i)6-s + (−1.46 − 2.54i)7-s − 6.31·8-s + (0.817 + 1.41i)9-s + (−4.45 + 3.54i)10-s + (−0.550 − 0.317i)11-s − 9.64i·12-s − 7.48·14-s + (−3.00 − 3.76i)15-s + (−3.55 + 6.16i)16-s + (−1.05 + 0.611i)17-s + 4.16·18-s + ⋯
L(s)  = 1  + (0.900 − 1.55i)2-s + (1.07 + 0.621i)3-s + (−1.12 − 1.94i)4-s + (−0.930 − 0.365i)5-s + (1.93 − 1.11i)6-s + (−0.555 − 0.961i)7-s − 2.23·8-s + (0.272 + 0.472i)9-s + (−1.40 + 1.12i)10-s + (−0.165 − 0.0957i)11-s − 2.78i·12-s − 1.99·14-s + (−0.774 − 0.972i)15-s + (−0.889 + 1.54i)16-s + (−0.257 + 0.148i)17-s + 0.981·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.992 - 0.118i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (654, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.992 - 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.133190 + 2.23758i\)
\(L(\frac12)\) \(\approx\) \(0.133190 + 2.23758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.08 + 0.817i)T \)
13 \( 1 \)
good2 \( 1 + (-1.27 + 2.20i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.86 - 1.07i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.46 + 2.54i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.550 + 0.317i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.05 - 0.611i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.18 - 0.682i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.86 - 1.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.96iT - 31T^{2} \)
37 \( 1 + (-0.611 + 1.05i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.62 - 4.98i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.18 + 0.683i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.16T + 47T^{2} \)
53 \( 1 - 0.642iT - 53T^{2} \)
59 \( 1 + (-6.57 + 3.79i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.13 - 1.96i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.01 - 6.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.28 + 1.31i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.3T + 73T^{2} \)
79 \( 1 + 1.03T + 79T^{2} \)
83 \( 1 + 11.8T + 83T^{2} \)
89 \( 1 + (-10.8 - 6.27i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.39 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.882586412764667290446113933791, −9.295145431330894467085758623645, −8.355296793399439967948838794808, −7.36061216838003230353173009263, −5.87287440943207305867996949722, −4.44492754440959686116134253260, −4.10120175345832340672153378065, −3.35922634215722505416167791525, −2.46227185404692568996120760789, −0.71990985327876468807973299900, 2.64137597025380075506618496531, 3.34150642649335608335672487214, 4.45136416334466097174267139050, 5.49899370080824991733529016790, 6.55408502883613996132515697633, 7.17993630242353814953316699186, 7.82246449518446624959246823050, 8.746287699415365791409072904690, 8.954462669138141750378349256559, 10.65360680585289365936281180371

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