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Properties

Label 2-845-65.32-c1-0-43
Degree $2$
Conductor $845$
Sign $-0.333 + 0.942i$
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  + (−0.246 + 0.427i)2-s + (−0.243 − 0.908i)3-s + (0.878 + 1.52i)4-s + (−2.21 + 0.284i)5-s + (0.448 + 0.120i)6-s + (−3.18 + 1.83i)7-s − 1.85·8-s + (1.83 − 1.05i)9-s + (0.426 − 1.01i)10-s + (0.664 − 0.177i)11-s + (1.16 − 1.16i)12-s − 1.81i·14-s + (0.798 + 1.94i)15-s + (−1.29 + 2.24i)16-s + (−2.29 − 0.614i)17-s + 1.04i·18-s + ⋯
L(s)  = 1  + (−0.174 + 0.302i)2-s + (−0.140 − 0.524i)3-s + (0.439 + 0.760i)4-s + (−0.991 + 0.127i)5-s + (0.183 + 0.0490i)6-s + (−1.20 + 0.694i)7-s − 0.655·8-s + (0.610 − 0.352i)9-s + (0.134 − 0.322i)10-s + (0.200 − 0.0536i)11-s + (0.337 − 0.337i)12-s − 0.485i·14-s + (0.206 + 0.502i)15-s + (−0.324 + 0.562i)16-s + (−0.556 − 0.149i)17-s + 0.246i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.333 + 0.942i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.333 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.192169 - 0.271900i\)
\(L(\frac12)\) \(\approx\) \(0.192169 - 0.271900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.21 - 0.284i)T \)
13 \( 1 \)
good2 \( 1 + (0.246 - 0.427i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.243 + 0.908i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (3.18 - 1.83i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.664 + 0.177i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.29 + 0.614i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.41 + 5.29i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.30 - 0.350i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (8.24 + 4.75i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.81 + 4.81i)T - 31iT^{2} \)
37 \( 1 + (1.58 + 0.917i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.143 - 0.534i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.560 + 2.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + 3.80iT - 47T^{2} \)
53 \( 1 + (2.47 - 2.47i)T - 53iT^{2} \)
59 \( 1 + (10.0 + 2.69i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.09 + 5.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.12 - 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.47 - 1.73i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 - 3.37T + 73T^{2} \)
79 \( 1 - 3.12iT - 79T^{2} \)
83 \( 1 + 2.13iT - 83T^{2} \)
89 \( 1 + (-0.874 - 3.26i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.53 + 6.12i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.640840637464054994572142916598, −9.052831113282643803220244133445, −8.050145642323936342704445287019, −7.25313428617894357153381029036, −6.70798643524016893761895783413, −5.96034902034883923335286917610, −4.29141484449157275650790272621, −3.39294350653949704448134200566, −2.41671568761304740805151693932, −0.17421643556788183511488700842, 1.46160846735637784424877193545, 3.20929261794699247055866733577, 3.98849099222660148341990291921, 5.01527158154313386030751701369, 6.20811918634472345703347149208, 6.99697699062574340610920009672, 7.78672505957026000775408970782, 9.112624090862479994726863979768, 9.751298061257776683999260502890, 10.56620972087949099081357707931

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