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Properties

Label 1-4033-4033.2092-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.720 + 0.693i$
Analytic cond. $433.406$
Root an. cond. $433.406$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + (−0.597 + 0.802i)3-s − 4-s + (−0.230 − 0.973i)5-s + (−0.802 − 0.597i)6-s + (−0.286 + 0.957i)7-s − i·8-s + (−0.286 − 0.957i)9-s + (0.973 − 0.230i)10-s + (−0.973 − 0.230i)11-s + (0.597 − 0.802i)12-s + (−0.230 − 0.973i)13-s + (−0.957 − 0.286i)14-s + (0.918 + 0.396i)15-s + 16-s + (0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + i·2-s + (−0.597 + 0.802i)3-s − 4-s + (−0.230 − 0.973i)5-s + (−0.802 − 0.597i)6-s + (−0.286 + 0.957i)7-s − i·8-s + (−0.286 − 0.957i)9-s + (0.973 − 0.230i)10-s + (−0.973 − 0.230i)11-s + (0.597 − 0.802i)12-s + (−0.230 − 0.973i)13-s + (−0.957 − 0.286i)14-s + (0.918 + 0.396i)15-s + 16-s + (0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.720 + 0.693i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2092, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.720 + 0.693i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1251117073 + 0.3101785019i\)
\(L(\frac12)\) \(\approx\) \(0.1251117073 + 0.3101785019i\)
\(L(1)\) \(\approx\) \(0.4672892875 + 0.3011224033i\)
\(L(1)\) \(\approx\) \(0.4672892875 + 0.3011224033i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (-0.597 + 0.802i)T \)
5 \( 1 + (-0.230 - 0.973i)T \)
7 \( 1 + (-0.286 + 0.957i)T \)
11 \( 1 + (-0.973 - 0.230i)T \)
13 \( 1 + (-0.230 - 0.973i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
19 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (-0.984 - 0.173i)T \)
29 \( 1 + (-0.957 - 0.286i)T \)
31 \( 1 + (0.448 + 0.893i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (0.597 + 0.802i)T \)
53 \( 1 + (-0.993 - 0.116i)T \)
59 \( 1 + (-0.918 - 0.396i)T \)
61 \( 1 + (-0.549 - 0.835i)T \)
67 \( 1 + (-0.597 - 0.802i)T \)
71 \( 1 + T \)
73 \( 1 + (0.286 + 0.957i)T \)
79 \( 1 + (-0.918 - 0.396i)T \)
83 \( 1 + (-0.0581 - 0.998i)T \)
89 \( 1 + (0.727 - 0.686i)T \)
97 \( 1 + (0.448 - 0.893i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.327814452100062133455660947720, −17.60464515397907380679057372536, −16.88712653933003773081268549927, −16.249231362259881861717253591687, −15.19254913568674741541507730905, −14.26727608315257181060100698458, −13.64588375987066662018656514158, −13.38413871970165065403179692515, −12.39170946866318067067965871241, −11.75697263330274980466929681131, −11.24759324168967127449969332058, −10.56920177988603421717276070669, −10.030670674765294196238439888682, −9.32318439020917110956658272969, −7.99582303315081362850799258397, −7.55852949182860676208231198797, −6.94702357960788013158232020884, −6.05032115074093485455375219467, −5.17887274129476710968366772148, −4.395459311920567147493089566418, −3.549888775633282008024993914910, −2.7007839959347440371882081275, −2.09735931522592035621758048743, −1.120636294398459098862618321571, −0.15932468526001034663593751671, 0.33008682723709989142948864826, 1.51190519172251158225295780047, 3.09156868850280283915063461867, 3.67324158796627540260213451492, 4.68929621570558522341862301623, 5.23338474449163692270692682323, 5.832507165020590605163936966091, 6.12068637062723101004771353669, 7.61329484654505102556340981183, 8.09341149781323182033302478917, 8.7051758095218863544733191585, 9.59953978943964474635854834925, 9.972646894084608761664010208113, 10.80566579578056737422458994443, 12.00192878294888539395354869935, 12.48482176842583043438445230997, 12.87791327987458179441474451105, 13.99013375209495811126189466343, 14.73943561191264460596075840927, 15.548926219004532138790620049389, 15.79084159731463983179623557399, 16.35114250023186007386217717355, 17.0492160358273559791129856417, 17.61697279864940304805696479609, 18.4195171192769069831267343697

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