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Properties

Label 2-845-65.33-c1-0-51
Degree $2$
Conductor $845$
Sign $-0.982 + 0.188i$
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  + (1.04 − 1.80i)2-s + (−2.66 + 0.713i)3-s + (−1.17 − 2.04i)4-s + (0.194 − 2.22i)5-s + (−1.48 + 5.55i)6-s + (2.52 − 1.45i)7-s − 0.750·8-s + (3.97 − 2.29i)9-s + (−3.82 − 2.67i)10-s + (−0.00681 − 0.0254i)11-s + (4.59 + 4.59i)12-s − 6.07i·14-s + (1.07 + 6.06i)15-s + (1.57 − 2.72i)16-s + (0.741 − 2.76i)17-s − 9.58i·18-s + ⋯
L(s)  = 1  + (0.738 − 1.27i)2-s + (−1.53 + 0.411i)3-s + (−0.589 − 1.02i)4-s + (0.0869 − 0.996i)5-s + (−0.607 + 2.26i)6-s + (0.952 − 0.550i)7-s − 0.265·8-s + (1.32 − 0.765i)9-s + (−1.20 − 0.846i)10-s + (−0.00205 − 0.00767i)11-s + (1.32 + 1.32i)12-s − 1.62i·14-s + (0.276 + 1.56i)15-s + (0.393 − 0.682i)16-s + (0.179 − 0.671i)17-s − 2.26i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.982 + 0.188i$
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: \((2,\ 845,\ (\ :1/2),\ -0.982 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.139550 - 1.46593i\)
\(L(\frac12)\) \(\approx\) \(0.139550 - 1.46593i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.194 + 2.22i)T \)
13 \( 1 \)
good2 \( 1 + (-1.04 + 1.80i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (2.66 - 0.713i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (-2.52 + 1.45i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.00681 + 0.0254i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.741 + 2.76i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.62 - 1.23i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.0961 - 0.358i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (3.62 + 2.09i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.835 - 0.835i)T + 31iT^{2} \)
37 \( 1 + (5.58 + 3.22i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.57 - 2.02i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.69 + 1.79i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 - 0.833iT - 47T^{2} \)
53 \( 1 + (-0.902 - 0.902i)T + 53iT^{2} \)
59 \( 1 + (-0.387 + 1.44i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.35 - 9.26i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.15 + 10.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.957 - 3.57i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 - 4.25iT - 79T^{2} \)
83 \( 1 + 1.31iT - 83T^{2} \)
89 \( 1 + (-3.23 + 0.867i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.202 + 0.351i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.12396152588068723184453496031, −9.525129233199856135317728758724, −8.134397019117011008996755886374, −7.07990822053778686742934440830, −5.59381903757429464480084599084, −5.11861677262200879823353539014, −4.52513655629024366514929673503, −3.59098750789772594676119014552, −1.72583650053081802447716346691, −0.74365105386565660617043900097, 1.72457389597469940919036119248, 3.62620243539026478638941094185, 5.08526131226655922350530692269, 5.30842565545677218982405132661, 6.28014246801258990863146495117, 6.83200386868537067915819513993, 7.57582617657309938835024766050, 8.427859721609985518358514504090, 9.999101447010529260516105842799, 10.82858445715184854880469638611

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