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Properties

Label 2-845-13.12-c1-0-14
Degree $2$
Conductor $845$
Sign $-0.722 + 0.691i$
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  + 2.63i·2-s + 1.98·3-s − 4.94·4-s + i·5-s + 5.24i·6-s + 3.28i·7-s − 7.77i·8-s + 0.957·9-s − 2.63·10-s + 3.22i·11-s − 9.84·12-s − 8.65·14-s + 1.98i·15-s + 10.5·16-s + 4.25·17-s + 2.52i·18-s + ⋯
L(s)  = 1  + 1.86i·2-s + 1.14·3-s − 2.47·4-s + 0.447i·5-s + 2.14i·6-s + 1.24i·7-s − 2.74i·8-s + 0.319·9-s − 0.833·10-s + 0.973i·11-s − 2.84·12-s − 2.31·14-s + 0.513i·15-s + 2.64·16-s + 1.03·17-s + 0.595i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.722 + 0.691i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (506, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.624111 - 1.55390i\)
\(L(\frac12)\) \(\approx\) \(0.624111 - 1.55390i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - iT \)
13 \( 1 \)
good2 \( 1 - 2.63iT - 2T^{2} \)
3 \( 1 - 1.98T + 3T^{2} \)
7 \( 1 - 3.28iT - 7T^{2} \)
11 \( 1 - 3.22iT - 11T^{2} \)
17 \( 1 - 4.25T + 17T^{2} \)
19 \( 1 + 2.87iT - 19T^{2} \)
23 \( 1 + 6.09T + 23T^{2} \)
29 \( 1 - 5.77T + 29T^{2} \)
31 \( 1 + 0.835iT - 31T^{2} \)
37 \( 1 - 5.59iT - 37T^{2} \)
41 \( 1 - 2.18iT - 41T^{2} \)
43 \( 1 + 2.48T + 43T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 + 5.08T + 53T^{2} \)
59 \( 1 - 0.144iT - 59T^{2} \)
61 \( 1 - 6.06T + 61T^{2} \)
67 \( 1 - 12.6iT - 67T^{2} \)
71 \( 1 + 9.02iT - 71T^{2} \)
73 \( 1 - 5.70iT - 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 - 7.41iT - 83T^{2} \)
89 \( 1 - 13.0iT - 89T^{2} \)
97 \( 1 - 2.97iT - 97T^{2} \)
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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

−9.982206353435981059380237509868, −9.578875051851724569759135387602, −8.590276021878866773501462215789, −8.208523632403104196148788558548, −7.38913467138675937602651421011, −6.52558509710811261323743740108, −5.66282667804667593288254293137, −4.77373559910209486297597294481, −3.55992732279227890503077466659, −2.34903168027664040392753990977, 0.73686898895001015142228184997, 1.89513597880046272074423992635, 3.18706443470151577878021671407, 3.67294960364755747137579458301, 4.55187895371996325882185025831, 5.86726357326031116774101483704, 7.73590910033524672610570471647, 8.240131912406912984138900588014, 9.059520817888694699954589037626, 9.869298979934844668990214373915

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