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Properties

Label 1-4033-4033.2341-r1-0-0
Degree $1$
Conductor $4033$
Sign $-0.803 - 0.595i$
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  + (0.342 + 0.939i)2-s + (−0.993 + 0.116i)3-s + (−0.766 + 0.642i)4-s + (−0.286 − 0.957i)5-s + (−0.448 − 0.893i)6-s + (−0.686 + 0.727i)7-s + (−0.866 − 0.5i)8-s + (0.973 − 0.230i)9-s + (0.802 − 0.597i)10-s + (0.802 + 0.597i)11-s + (0.686 − 0.727i)12-s + (−0.727 − 0.686i)13-s + (−0.918 − 0.396i)14-s + (0.396 + 0.918i)15-s + (0.173 − 0.984i)16-s + (−0.984 − 0.173i)17-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (−0.993 + 0.116i)3-s + (−0.766 + 0.642i)4-s + (−0.286 − 0.957i)5-s + (−0.448 − 0.893i)6-s + (−0.686 + 0.727i)7-s + (−0.866 − 0.5i)8-s + (0.973 − 0.230i)9-s + (0.802 − 0.597i)10-s + (0.802 + 0.597i)11-s + (0.686 − 0.727i)12-s + (−0.727 − 0.686i)13-s + (−0.918 − 0.396i)14-s + (0.396 + 0.918i)15-s + (0.173 − 0.984i)16-s + (−0.984 − 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.803 - 0.595i$
Analytic conductor: \(433.406\)
Root analytic conductor: \(433.406\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (2341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (1:\ ),\ -0.803 - 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1023203124 + 0.3098178613i\)
\(L(\frac12)\) \(\approx\) \(-0.1023203124 + 0.3098178613i\)
\(L(1)\) \(\approx\) \(0.5843092729 + 0.3184698072i\)
\(L(1)\) \(\approx\) \(0.5843092729 + 0.3184698072i\)

Euler product

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

−18.16316263072325818760219519147, −17.28829991311813930871419949029, −16.64840088480056494364894700019, −15.911014296249502023609210522288, −15.078169357877180009612718061144, −14.16752140488042262118730579683, −13.817022466644771023502159542099, −12.9739730976902899687969754161, −12.18505343588756128904742340939, −11.68365169573008953631797480322, −11.00343896512849850620232015542, −10.59722960747048454106854432181, −9.78515601705256498520318627309, −9.29810005838015586069411854312, −8.08371611128112682482104298225, −6.93603192703231999670668540465, −6.64251504689324963887055213804, −5.95384477036418727392821299120, −4.88562846221851843271312224230, −4.19770965394080496015397808398, −3.61503869403280039909362511422, −2.74266433366413473346125389256, −1.82931708577129259029617757017, −0.805231727681833164648795006860, −0.09133957092259915055664365937, 0.68308491708266139551171519199, 1.84429555848803759523423609830, 3.2300165661578956765078514692, 3.98639119319917238944434564052, 4.83620516236698638559116778500, 5.23707466010087105518655472101, 5.94325966850363305421641330337, 6.678710898537942727396486404333, 7.34543786113293126744705437512, 8.09957129243763539152473070051, 9.211037822397225389882513918795, 9.388278786082737749089615542, 10.21699576432427567617397921732, 11.67193188432616375941728896272, 11.89013100592536419767842817308, 12.658180387878556136018720222677, 13.015257712117946375017901301492, 13.90084491877708456554156243525, 15.00860065853264212114607305129, 15.42788729687683680265163261058, 16.13624839272101477758941294317, 16.41944660721027745898727830754, 17.345223847187004711724351107238, 17.74819315776436181828092791435, 18.27381750885006699863887072267

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