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Properties

Label 2-845-65.64-c1-0-54
Degree $2$
Conductor $845$
Sign $0.997 - 0.0752i$
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  + 2.18·2-s + i·3-s + 2.79·4-s + (0.456 − 2.18i)5-s + 2.18i·6-s + 1.73·7-s + 1.73·8-s + 2·9-s + (0.999 − 4.79i)10-s + 2.64i·11-s + 2.79i·12-s + 3.79·14-s + (2.18 + 0.456i)15-s − 1.79·16-s − 4.58i·17-s + 4.37·18-s + ⋯
L(s)  = 1  + 1.54·2-s + 0.577i·3-s + 1.39·4-s + (0.204 − 0.978i)5-s + 0.893i·6-s + 0.654·7-s + 0.612·8-s + 0.666·9-s + (0.316 − 1.51i)10-s + 0.797i·11-s + 0.805i·12-s + 1.01·14-s + (0.565 + 0.117i)15-s − 0.447·16-s − 1.11i·17-s + 1.03·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $0.997 - 0.0752i$
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: \((2,\ 845,\ (\ :1/2),\ 0.997 - 0.0752i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.99852 + 0.150568i\)
\(L(\frac12)\) \(\approx\) \(3.99852 + 0.150568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.456 + 2.18i)T \)
13 \( 1 \)
good2 \( 1 - 2.18T + 2T^{2} \)
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 \( 1 - 9.66iT - 31T^{2} \)
37 \( 1 - 7.93T + 37T^{2} \)
41 \( 1 + 2.64iT - 41T^{2} \)
43 \( 1 - 1.41iT - 43T^{2} \)
47 \( 1 + 8.75T + 47T^{2} \)
53 \( 1 + 1.58iT - 53T^{2} \)
59 \( 1 - 3.36iT - 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + 14.8T + 67T^{2} \)
71 \( 1 - 3.55iT - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 - 4.28iT - 89T^{2} \)
97 \( 1 + 4.47T + 97T^{2} \)
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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

−10.26935888590220180462032243625, −9.475767019135543653019147724552, −8.564242538192335328684141625549, −7.41544764977497919763257803137, −6.46760855518554272081149881727, −5.31664282778341715879067456877, −4.60928880668393207082648012070, −4.41198391430534985387751424916, −2.98971466191072346683982074287, −1.61171354484017049942117593865, 1.73135533685063801699589762900, 2.83953270391628201813491860056, 3.83851297687756267802733494150, 4.73641885061742317044847903636, 6.03014324078757055367899266680, 6.28221896131901469227806397818, 7.39489552679788987323149996768, 8.077665114388552060383125376690, 9.487043558633141272905636205979, 10.54972734100461153577907441393

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