↑

Properties

Label 1-3307-3307.1023-r1-0-0
Degree $1$
Conductor $3307$
Sign $0.424 - 0.905i$
Analytic cond. $355.386$
Root an. cond. $355.386$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.444 − 0.895i)2-s + (0.920 − 0.390i)3-s + (−0.604 + 0.796i)4-s + (0.595 + 0.803i)5-s + (−0.759 − 0.651i)6-s + (−0.990 − 0.134i)7-s + (0.982 + 0.187i)8-s + (0.695 − 0.718i)9-s + (0.454 − 0.890i)10-s + (−0.917 + 0.397i)11-s + (−0.245 + 0.969i)12-s + (0.961 + 0.275i)13-s + (0.320 + 0.947i)14-s + (0.861 + 0.507i)15-s + (−0.269 − 0.963i)16-s + (0.811 − 0.583i)17-s + ⋯
L(s)  = 1  + (−0.444 − 0.895i)2-s + (0.920 − 0.390i)3-s + (−0.604 + 0.796i)4-s + (0.595 + 0.803i)5-s + (−0.759 − 0.651i)6-s + (−0.990 − 0.134i)7-s + (0.982 + 0.187i)8-s + (0.695 − 0.718i)9-s + (0.454 − 0.890i)10-s + (−0.917 + 0.397i)11-s + (−0.245 + 0.969i)12-s + (0.961 + 0.275i)13-s + (0.320 + 0.947i)14-s + (0.861 + 0.507i)15-s + (−0.269 − 0.963i)16-s + (0.811 − 0.583i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3307 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3307 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3307\)
Sign: $0.424 - 0.905i$
Analytic conductor: \(355.386\)
Root analytic conductor: \(355.386\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3307} (1023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3307,\ (1:\ ),\ 0.424 - 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.994712552 - 1.267238914i\)
\(L(\frac12)\) \(\approx\) \(1.994712552 - 1.267238914i\)
\(L(1)\) \(\approx\) \(1.056313904 - 0.4272519350i\)
\(L(1)\) \(\approx\) \(1.056313904 - 0.4272519350i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3307 \( 1 \)
good2 \( 1 + (-0.444 - 0.895i)T \)
3 \( 1 + (0.920 - 0.390i)T \)
5 \( 1 + (0.595 + 0.803i)T \)
7 \( 1 + (-0.990 - 0.134i)T \)
11 \( 1 + (-0.917 + 0.397i)T \)
13 \( 1 + (0.961 + 0.275i)T \)
17 \( 1 + (0.811 - 0.583i)T \)
19 \( 1 + (-0.122 - 0.992i)T \)
23 \( 1 + (-0.905 - 0.423i)T \)
29 \( 1 + (0.553 + 0.832i)T \)
31 \( 1 + (-0.258 + 0.966i)T \)
37 \( 1 + (0.693 - 0.720i)T \)
41 \( 1 + (-0.957 + 0.288i)T \)
43 \( 1 + (-0.797 + 0.603i)T \)
47 \( 1 + (0.942 + 0.333i)T \)
53 \( 1 + (0.997 - 0.0664i)T \)
59 \( 1 + (0.749 - 0.662i)T \)
61 \( 1 + (0.931 + 0.363i)T \)
67 \( 1 + (-0.885 + 0.464i)T \)
71 \( 1 + (-0.265 - 0.964i)T \)
73 \( 1 + (-0.841 - 0.539i)T \)
79 \( 1 + (-0.312 + 0.949i)T \)
83 \( 1 + (0.666 - 0.745i)T \)
89 \( 1 + (0.459 - 0.887i)T \)
97 \( 1 + (0.278 - 0.960i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.7672551800188036606941655787, −18.27539175934779468355384147863, −17.22626323557100060897557680167, −16.410416572507216073058315021143, −16.22450025782594417366628392947, −15.4496622608776210956915310967, −14.88315563103226366149427742079, −13.81943706174828153849730191341, −13.44111572599730461242715586671, −13.02136383620135549790289258141, −11.98570558694677837883827380520, −10.39191023878066718002992455729, −10.18951732928961709456729463210, −9.57944247497326460019327336843, −8.71392650962582130922824487447, −8.232860605286822776649973309031, −7.74944657080599641112766194964, −6.56037659848490217929296931005, −5.68759832163974022109482962323, −5.499269817538811858667363243893, −4.16407433833457412479530944658, −3.664642331752358241916127371254, −2.49870200229035496036051390525, −1.577743640079466423098676747149, −0.62489334962753030934023799131, 0.5474967212032814282516446764, 1.54072533386921418512952478694, 2.39573104874911373712684821687, 2.9836869899659744560950106845, 3.45303419813591118301484354906, 4.42513576930516346630656471283, 5.60815073734963759494323459154, 6.72527248996055787336106526794, 7.13658721202487356285478583984, 8.00441324448406441010580386767, 8.844830639368037197502927675724, 9.4089547157395063749115446162, 10.244036752218214097816644903825, 10.42943307243102780123901300949, 11.54120525516749884714293762649, 12.400701112210836117522809647849, 13.08449362267461658502428562204, 13.53458349507751145976635915295, 14.1072376094949987114823257606, 14.93270900245359618733507098350, 15.9829965096256277892756169415, 16.37344736010280842695259110197, 17.74768627569168705695252192313, 18.0522470737150015323965305003, 18.67153784884567133424465959864

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