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Properties

Label 1-647-647.468-r0-0-0
Degree $1$
Conductor $647$
Sign $0.235 - 0.971i$
Analytic cond. $3.00465$
Root an. cond. $3.00465$
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  + (−0.273 − 0.961i)2-s + (−0.982 + 0.183i)3-s + (−0.850 + 0.526i)4-s + (0.932 − 0.361i)5-s + (0.445 + 0.895i)6-s + (0.739 − 0.673i)7-s + (0.739 + 0.673i)8-s + (0.932 − 0.361i)9-s + (−0.602 − 0.798i)10-s + (0.932 + 0.361i)11-s + (0.739 − 0.673i)12-s + (0.739 − 0.673i)13-s + (−0.850 − 0.526i)14-s + (−0.850 + 0.526i)15-s + (0.445 − 0.895i)16-s + (−0.982 + 0.183i)17-s + ⋯
L(s)  = 1  + (−0.273 − 0.961i)2-s + (−0.982 + 0.183i)3-s + (−0.850 + 0.526i)4-s + (0.932 − 0.361i)5-s + (0.445 + 0.895i)6-s + (0.739 − 0.673i)7-s + (0.739 + 0.673i)8-s + (0.932 − 0.361i)9-s + (−0.602 − 0.798i)10-s + (0.932 + 0.361i)11-s + (0.739 − 0.673i)12-s + (0.739 − 0.673i)13-s + (−0.850 − 0.526i)14-s + (−0.850 + 0.526i)15-s + (0.445 − 0.895i)16-s + (−0.982 + 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(647\)
Sign: $0.235 - 0.971i$
Analytic conductor: \(3.00465\)
Root analytic conductor: \(3.00465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{647} (468, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 647,\ (0:\ ),\ 0.235 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9538057472 - 0.7505181950i\)
\(L(\frac12)\) \(\approx\) \(0.9538057472 - 0.7505181950i\)
\(L(1)\) \(\approx\) \(0.8301948504 - 0.4318852611i\)
\(L(1)\) \(\approx\) \(0.8301948504 - 0.4318852611i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad647 \( 1 \)
good2 \( 1 + (-0.273 - 0.961i)T \)
3 \( 1 + (-0.982 + 0.183i)T \)
5 \( 1 + (0.932 - 0.361i)T \)
7 \( 1 + (0.739 - 0.673i)T \)
11 \( 1 + (0.932 + 0.361i)T \)
13 \( 1 + (0.739 - 0.673i)T \)
17 \( 1 + (-0.982 + 0.183i)T \)
19 \( 1 + (0.445 - 0.895i)T \)
23 \( 1 + (0.445 + 0.895i)T \)
29 \( 1 + (0.445 + 0.895i)T \)
31 \( 1 + (0.932 + 0.361i)T \)
37 \( 1 + (0.445 + 0.895i)T \)
41 \( 1 + (0.739 + 0.673i)T \)
43 \( 1 + (-0.273 + 0.961i)T \)
47 \( 1 + (-0.982 + 0.183i)T \)
53 \( 1 + (0.445 + 0.895i)T \)
59 \( 1 + (-0.602 - 0.798i)T \)
61 \( 1 + (-0.982 - 0.183i)T \)
67 \( 1 + (-0.982 - 0.183i)T \)
71 \( 1 + (0.445 + 0.895i)T \)
73 \( 1 + (-0.273 - 0.961i)T \)
79 \( 1 + (-0.273 + 0.961i)T \)
83 \( 1 + (-0.602 + 0.798i)T \)
89 \( 1 + (-0.273 - 0.961i)T \)
97 \( 1 + (-0.602 - 0.798i)T \)
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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

−22.88796720173381676375368240318, −22.52981952223076103347019339005, −21.62090682080977963703099845804, −20.97089660776065324225185543946, −19.27250468852321494935964699315, −18.50809442762905733094823183484, −18.01138029998821242147502165481, −17.28037898140079875679347460557, −16.61256567867820134945915415157, −15.76079405351856561595013448737, −14.75944164080138575336831301451, −13.97645043777203095647730595120, −13.27413116048178761254669830705, −12.03239830841510984773820980813, −11.16948391964887664143906344025, −10.30567895811959258223658968490, −9.26507436302705134355863040357, −8.57286308343290729973753177337, −7.322847819837982681815631949867, −6.20223151116080620869231487463, −6.15059964232056954390342934360, −4.99389908630130374575039214448, −4.114286647688145230445681232268, −2.07871386096350978487089228887, −1.11109766986759098217343235589, 1.11041833628067393812630690947, 1.486345332862148000806613872499, 3.10584771468887868621134610324, 4.49277510388958956974298190804, 4.83362032526332535122447198065, 6.090451578679313380320042088838, 7.11712728753417284083750485274, 8.42767026569487557813614370189, 9.36828094925705827619949406419, 10.08467521966849569454757460199, 11.02659281245271074240496712533, 11.41848199790051180941381672071, 12.56013978326988480706069476331, 13.30266617938231096244775666337, 13.98501578181465159789180622247, 15.26794632926865471516776091955, 16.52397695540153701771282181916, 17.26398638730523721652625105914, 17.80630088606901901229205621360, 18.15267443245457236509641217017, 19.71044018177471461516179154637, 20.25422480726416822386111984641, 21.224752921522227142753766084969, 21.68255805167794310069063903965, 22.53603876468071180756208903912

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