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Properties

Label 2-845-13.3-c1-0-17
Degree $2$
Conductor $845$
Sign $0.872 - 0.488i$
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  + (0.866 + 1.5i)2-s + (−1.36 − 2.36i)3-s + (−0.5 + 0.866i)4-s − 5-s + (2.36 − 4.09i)6-s + (−1 + 1.73i)7-s + 1.73·8-s + (−2.23 + 3.86i)9-s + (−0.866 − 1.5i)10-s + (2.36 + 4.09i)11-s + 2.73·12-s − 3.46·14-s + (1.36 + 2.36i)15-s + (2.49 + 4.33i)16-s + (1.73 − 3i)17-s − 7.73·18-s + ⋯
L(s)  = 1  + (0.612 + 1.06i)2-s + (−0.788 − 1.36i)3-s + (−0.250 + 0.433i)4-s − 0.447·5-s + (0.965 − 1.67i)6-s + (−0.377 + 0.654i)7-s + 0.612·8-s + (−0.744 + 1.28i)9-s + (−0.273 − 0.474i)10-s + (0.713 + 1.23i)11-s + 0.788·12-s − 0.925·14-s + (0.352 + 0.610i)15-s + (0.624 + 1.08i)16-s + (0.420 − 0.727i)17-s − 1.82·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.872 - 0.488i$
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: \((2,\ 845,\ (\ :1/2),\ 0.872 - 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61063 + 0.420518i\)
\(L(\frac12)\) \(\approx\) \(1.61063 + 0.420518i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.36 + 2.36i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.36 - 4.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.73 + 3i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.09 + 5.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.633 + 1.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.26 - 2.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.73 + 3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0980 + 0.169i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + (4.56 - 7.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.19 + 7.26i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.19 + 5.53i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.36 - 4.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 8.39T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-6.46 - 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.33153772746280101497876164724, −9.225633125648916399421118116931, −8.083672068505606576985172912494, −7.14468848013609227806312526978, −6.93267925528450768554711475547, −6.08904659888843860045667431676, −5.22325180445996224570907537604, −4.40644514780993295445718047651, −2.60628266093576632095115238579, −1.12029850915767461301154858460, 0.994953194282242462011458313859, 3.11373656333181038447660277261, 3.85696348823392790327807916544, 4.25011085202845757737911249053, 5.46178755988444512437043212557, 6.24146356460353384143682324615, 7.62583907849728397417550352022, 8.666461070035397363708566544893, 9.968942144487184590624042115876, 10.21155197533640190177558938569

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