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Properties

Label 2-845-13.3-c1-0-43
Degree $2$
Conductor $845$
Sign $-0.945 + 0.326i$
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  + (−0.747 − 1.29i)2-s + (−0.0473 − 0.0820i)3-s + (−0.118 + 0.204i)4-s + 5-s + (−0.0708 + 0.122i)6-s + (2.41 − 4.18i)7-s − 2.63·8-s + (1.49 − 2.59i)9-s + (−0.747 − 1.29i)10-s + (−0.534 − 0.926i)11-s + 0.0224·12-s − 7.21·14-s + (−0.0473 − 0.0820i)15-s + (2.20 + 3.82i)16-s + (−1.77 + 3.08i)17-s − 4.47·18-s + ⋯
L(s)  = 1  + (−0.528 − 0.915i)2-s + (−0.0273 − 0.0473i)3-s + (−0.0591 + 0.102i)4-s + 0.447·5-s + (−0.0289 + 0.0501i)6-s + (0.912 − 1.57i)7-s − 0.932·8-s + (0.498 − 0.863i)9-s + (−0.236 − 0.409i)10-s + (−0.161 − 0.279i)11-s + 0.00647·12-s − 1.92·14-s + (−0.0122 − 0.0211i)15-s + (0.552 + 0.956i)16-s + (−0.431 + 0.747i)17-s − 1.05·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.945 + 0.326i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.945 + 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.226953 - 1.35205i\)
\(L(\frac12)\) \(\approx\) \(0.226953 - 1.35205i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
13 \( 1 \)
good2 \( 1 + (0.747 + 1.29i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.0473 + 0.0820i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.41 + 4.18i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.534 + 0.926i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.77 - 3.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.86 + 4.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.54 - 6.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.736 + 1.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 + (0.0126 + 0.0219i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.133 - 0.232i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.77 - 3.08i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.51T + 47T^{2} \)
53 \( 1 - 0.991T + 53T^{2} \)
59 \( 1 + (-4.36 + 7.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.16 - 5.48i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.58 - 4.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.88 - 6.72i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 8.78T + 79T^{2} \)
83 \( 1 - 0.725T + 83T^{2} \)
89 \( 1 + (-6.75 - 11.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.71 - 2.97i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.885011139378536373738290564410, −9.326058466203797938420048307780, −8.302205944972082362720160285367, −7.23689921502422585531885153363, −6.52304717602297599711678249883, −5.29003017500586119776148896038, −4.14199261643933297261626175206, −3.15038477548417246592115623155, −1.63215417421461518845402725940, −0.851816672661626482220210886722, 1.91711729101855349514306535483, 2.86345210748665858378528249489, 4.78103976498932175872321903935, 5.39982188691544057403922058462, 6.28820165258194532029230110490, 7.29788315267811998727009419356, 8.069355063536038073311953202122, 8.714464574936282902891143256001, 9.422013955576008874731675919394, 10.38417841277629524222631421617

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