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Properties

Label 1-3407-3407.1022-r0-0-0
Degree $1$
Conductor $3407$
Sign $0.944 - 0.328i$
Analytic cond. $15.8220$
Root an. cond. $15.8220$
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  + (0.764 − 0.644i)2-s + (0.571 + 0.820i)3-s + (0.167 − 0.985i)4-s + (0.999 − 0.0368i)5-s + (0.965 + 0.258i)6-s + (−0.532 + 0.846i)7-s + (−0.507 − 0.861i)8-s + (−0.347 + 0.937i)9-s + (0.739 − 0.672i)10-s + (−0.455 + 0.890i)11-s + (0.905 − 0.425i)12-s + (0.558 − 0.829i)13-s + (0.138 + 0.990i)14-s + (0.600 + 0.799i)15-s + (−0.943 − 0.331i)16-s + (−0.392 − 0.919i)17-s + ⋯
L(s)  = 1  + (0.764 − 0.644i)2-s + (0.571 + 0.820i)3-s + (0.167 − 0.985i)4-s + (0.999 − 0.0368i)5-s + (0.965 + 0.258i)6-s + (−0.532 + 0.846i)7-s + (−0.507 − 0.861i)8-s + (−0.347 + 0.937i)9-s + (0.739 − 0.672i)10-s + (−0.455 + 0.890i)11-s + (0.905 − 0.425i)12-s + (0.558 − 0.829i)13-s + (0.138 + 0.990i)14-s + (0.600 + 0.799i)15-s + (−0.943 − 0.331i)16-s + (−0.392 − 0.919i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3407\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(15.8220\)
Root analytic conductor: \(15.8220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3407} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3407,\ (0:\ ),\ 0.944 - 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.818284941 - 0.6449254218i\)
\(L(\frac12)\) \(\approx\) \(3.818284941 - 0.6449254218i\)
\(L(1)\) \(\approx\) \(2.127932880 - 0.2701425043i\)
\(L(1)\) \(\approx\) \(2.127932880 - 0.2701425043i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3407 \( 1 \)
good2 \( 1 + (0.764 - 0.644i)T \)
3 \( 1 + (0.571 + 0.820i)T \)
5 \( 1 + (0.999 - 0.0368i)T \)
7 \( 1 + (-0.532 + 0.846i)T \)
11 \( 1 + (-0.455 + 0.890i)T \)
13 \( 1 + (0.558 - 0.829i)T \)
17 \( 1 + (-0.392 - 0.919i)T \)
19 \( 1 + (0.105 - 0.994i)T \)
23 \( 1 + (0.811 - 0.583i)T \)
29 \( 1 + (0.885 + 0.464i)T \)
31 \( 1 + (0.941 - 0.336i)T \)
37 \( 1 + (-0.988 + 0.152i)T \)
41 \( 1 + (0.272 + 0.962i)T \)
43 \( 1 + (0.906 - 0.421i)T \)
47 \( 1 + (0.901 - 0.431i)T \)
53 \( 1 + (0.407 - 0.913i)T \)
59 \( 1 + (0.414 + 0.910i)T \)
61 \( 1 + (-0.319 + 0.947i)T \)
67 \( 1 + (0.962 + 0.269i)T \)
71 \( 1 + (-0.266 + 0.963i)T \)
73 \( 1 + (0.768 - 0.639i)T \)
79 \( 1 + (0.954 - 0.297i)T \)
83 \( 1 + (-0.902 - 0.430i)T \)
89 \( 1 + (-0.497 + 0.867i)T \)
97 \( 1 + (-0.821 - 0.570i)T \)
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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

−18.92311331077366978521717579823, −18.02707901839890236456580611520, −17.1953639368307119051605739108, −16.98954250183551264465812603621, −15.9605006731308035933914001161, −15.39205201695811737816714564640, −14.130594302860540005483592365663, −14.02265202594346058165033404467, −13.569613139761748036245547780295, −12.78600643567802414577227349053, −12.348669042513103829634920392326, −11.2003552943712674978925348216, −10.52165673935934626339639182967, −9.4535770088956305844292657531, −8.72601974034878166346807786440, −8.11480606325453072570686787492, −7.26257676534524641286953420658, −6.50563489187143972016396700465, −6.17835460759886166086293626190, −5.41630773187997734242424180584, −4.207350580802883442632343199188, −3.52091969553045302565187695814, −2.82143294842499899554253952799, −1.929656800972036682544937935264, −0.997710983591824840753034357085, 0.923128197707941693314470566641, 2.28726065340385462831779772157, 2.61279241439196961509817450145, 3.1471744849895753807053218363, 4.33541786169795360952007940874, 5.08836241343013434009474260068, 5.4028066833064297302149085208, 6.40031439312648159983287728199, 7.13775540245641384908788292840, 8.59318275809968163406430806008, 9.05972924710564817284328956227, 9.79169438807081772566589432168, 10.28648372061665891834957755395, 10.91046242676908117261882436853, 11.8447310320245422620271707155, 12.73173697564114192185635263063, 13.25387271989433215667889727718, 13.779538786913667400701434017740, 14.5910893435089915111943775979, 15.38152967500137552265407806186, 15.58732763571758700051007227070, 16.41134879280542820201819799589, 17.59563476917199545210472216109, 18.152331365384807793125896776869, 18.90465842074046519462044652516

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