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Properties

Label 2-845-65.64-c1-0-37
Degree $2$
Conductor $845$
Sign $-0.467 + 0.883i$
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  − 0.456·2-s − i·3-s − 1.79·4-s + (−2.18 − 0.456i)5-s + 0.456i·6-s + 1.73·7-s + 1.73·8-s + 2·9-s + (0.999 + 0.208i)10-s + 2.64i·11-s + 1.79i·12-s − 0.791·14-s + (−0.456 + 2.18i)15-s + 2.79·16-s − 4.58i·17-s − 0.913·18-s + ⋯
L(s)  = 1  − 0.323·2-s − 0.577i·3-s − 0.895·4-s + (−0.978 − 0.204i)5-s + 0.186i·6-s + 0.654·7-s + 0.612·8-s + 0.666·9-s + (0.316 + 0.0660i)10-s + 0.797i·11-s + 0.517i·12-s − 0.211·14-s + (−0.117 + 0.565i)15-s + 0.697·16-s − 1.11i·17-s − 0.215·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.467 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.467 + 0.883i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (844, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.467 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.380501 - 0.631900i\)
\(L(\frac12)\) \(\approx\) \(0.380501 - 0.631900i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.18 + 0.456i)T \)
13 \( 1 \)
good2 \( 1 + 0.456T + 2T^{2} \)
3 \( 1 + iT - 3T^{2} \)
7 \( 1 - 1.73T + 7T^{2} \)
11 \( 1 - 2.64iT - 11T^{2} \)
17 \( 1 + 4.58iT - 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + 4.58iT - 23T^{2} \)
29 \( 1 + 4.58T + 29T^{2} \)
31 \( 1 - 6.20iT - 31T^{2} \)
37 \( 1 + 7.93T + 37T^{2} \)
41 \( 1 + 2.64iT - 41T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 - 1.82T + 47T^{2} \)
53 \( 1 + 7.58iT - 53T^{2} \)
59 \( 1 + 13.9iT - 59T^{2} \)
61 \( 1 + 1.41T + 61T^{2} \)
67 \( 1 - 1.00T + 67T^{2} \)
71 \( 1 - 7.02iT - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 6.01T + 83T^{2} \)
89 \( 1 + 9.57iT - 89T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−9.847825874722976715749555090859, −8.884674032366400508195316871249, −8.265896432191867165400459185010, −7.33931055106701086943905882985, −6.96518803156956725582251506532, −5.10405260713151354439581756043, −4.65202236975652413770558801387, −3.61801139786605981297865654346, −1.84202681435371838716556680459, −0.47324385330538669621027467174, 1.35788848751368419948665191004, 3.49190029078110399217059421361, 4.08259881604319400214909802589, 4.91071131023640204537900343018, 5.98625128878306903346023122883, 7.47296743019483784259357917245, 7.974919598591359704736419616734, 8.780686325854873679310278013679, 9.578872520482157668541498024401, 10.51023334581259651023194812666

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