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Properties

Label 1-348-348.35-r0-0-0
Degree $1$
Conductor $348$
Sign $0.357 + 0.934i$
Analytic cond. $1.61610$
Root an. cond. $1.61610$
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  + (0.222 + 0.974i)5-s + (0.900 + 0.433i)7-s + (−0.623 + 0.781i)11-s + (0.623 − 0.781i)13-s + 17-s + (−0.900 + 0.433i)19-s + (−0.222 + 0.974i)23-s + (−0.900 + 0.433i)25-s + (−0.222 − 0.974i)31-s + (−0.222 + 0.974i)35-s + (−0.623 − 0.781i)37-s + 41-s + (−0.222 + 0.974i)43-s + (−0.623 + 0.781i)47-s + (0.623 + 0.781i)49-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)5-s + (0.900 + 0.433i)7-s + (−0.623 + 0.781i)11-s + (0.623 − 0.781i)13-s + 17-s + (−0.900 + 0.433i)19-s + (−0.222 + 0.974i)23-s + (−0.900 + 0.433i)25-s + (−0.222 − 0.974i)31-s + (−0.222 + 0.974i)35-s + (−0.623 − 0.781i)37-s + 41-s + (−0.222 + 0.974i)43-s + (−0.623 + 0.781i)47-s + (0.623 + 0.781i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 348 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(348\)    =    \(2^{2} \cdot 3 \cdot 29\)
Sign: $0.357 + 0.934i$
Analytic conductor: \(1.61610\)
Root analytic conductor: \(1.61610\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{348} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 348,\ (0:\ ),\ 0.357 + 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.150135010 + 0.7914763729i\)
\(L(\frac12)\) \(\approx\) \(1.150135010 + 0.7914763729i\)
\(L(1)\) \(\approx\) \(1.123623975 + 0.3431650221i\)
\(L(1)\) \(\approx\) \(1.123623975 + 0.3431650221i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
29 \( 1 \)
good5 \( 1 + (0.222 + 0.974i)T \)
7 \( 1 + (0.900 + 0.433i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.900 + 0.433i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (-0.222 - 0.974i)T \)
37 \( 1 + (-0.623 - 0.781i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (0.222 + 0.974i)T \)
59 \( 1 + T \)
61 \( 1 + (0.900 + 0.433i)T \)
67 \( 1 + (-0.623 - 0.781i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 + (0.623 + 0.781i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (0.900 - 0.433i)T \)
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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

−24.483653382475935074119238881424, −23.83576212724302268234922604457, −23.28426574669310925106459965440, −21.71246860807957192734210152491, −21.0544903729172323437287861768, −20.558116783875809000520371572158, −19.359974845840759613781954509538, −18.449869692691272599183037621109, −17.444159662941652503700666109638, −16.6011942023627116733132163196, −15.98311449007552354674826953334, −14.60798884900217771299470700786, −13.8141925577164580787082826735, −12.97248388474508460281459921342, −11.921894811243631887788922829891, −10.96542214093676931739133182860, −10.03474054754239015895301333098, −8.56217668422006884323697776414, −8.35877270461123323427537082023, −6.900257286925355190505719989737, −5.62856237671025038218971702421, −4.784487411516396872665502227345, −3.75131805328360967658158580818, −2.09543566511618312747618937825, −0.9331004680318421614348499476, 1.67854328732022949752134278541, 2.70818213629056812731434835169, 3.93339303038606835622149378287, 5.334695258466713715706395722717, 6.0910831325349740132106153184, 7.52349857386619854524498234228, 8.04677810208888186639743672682, 9.497066549259106086763846521501, 10.46612863281658392515666520253, 11.168388654374213807837269143, 12.269751053198327670212511653373, 13.284248611195772745434868138108, 14.442888050312588065713627328303, 14.990696789130017884418832774705, 15.84557473174188578811930609349, 17.27945980779520027761881669585, 17.998933031110149682165701820872, 18.58711149377484274109639066538, 19.62555774737982152624729422963, 21.01713331118846014907333097850, 21.19101498542873750037953919017, 22.5365234060549943039679588012, 23.14259091103683652427787475419, 24.04408064403426653108910485934, 25.39614558439724914032107352840

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