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Properties

Label 1-344-344.181-r0-0-0
Degree $1$
Conductor $344$
Sign $-0.822 - 0.568i$
Analytic cond. $1.59752$
Root an. cond. $1.59752$
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  + (−0.955 − 0.294i)3-s + (−0.365 − 0.930i)5-s + (−0.5 − 0.866i)7-s + (0.826 + 0.563i)9-s + (0.900 − 0.433i)11-s + (0.988 + 0.149i)13-s + (0.0747 + 0.997i)15-s + (0.365 − 0.930i)17-s + (−0.826 + 0.563i)19-s + (0.222 + 0.974i)21-s + (0.0747 − 0.997i)23-s + (−0.733 + 0.680i)25-s + (−0.623 − 0.781i)27-s + (−0.955 + 0.294i)29-s + (−0.733 − 0.680i)31-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)3-s + (−0.365 − 0.930i)5-s + (−0.5 − 0.866i)7-s + (0.826 + 0.563i)9-s + (0.900 − 0.433i)11-s + (0.988 + 0.149i)13-s + (0.0747 + 0.997i)15-s + (0.365 − 0.930i)17-s + (−0.826 + 0.563i)19-s + (0.222 + 0.974i)21-s + (0.0747 − 0.997i)23-s + (−0.733 + 0.680i)25-s + (−0.623 − 0.781i)27-s + (−0.955 + 0.294i)29-s + (−0.733 − 0.680i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $-0.822 - 0.568i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ -0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1985815654 - 0.6372256025i\)
\(L(\frac12)\) \(\approx\) \(0.1985815654 - 0.6372256025i\)
\(L(1)\) \(\approx\) \(0.6222139329 - 0.3429925681i\)
\(L(1)\) \(\approx\) \(0.6222139329 - 0.3429925681i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (-0.955 - 0.294i)T \)
5 \( 1 + (-0.365 - 0.930i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (0.988 + 0.149i)T \)
17 \( 1 + (0.365 - 0.930i)T \)
19 \( 1 + (-0.826 + 0.563i)T \)
23 \( 1 + (0.0747 - 0.997i)T \)
29 \( 1 + (-0.955 + 0.294i)T \)
31 \( 1 + (-0.733 - 0.680i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
59 \( 1 + (-0.623 - 0.781i)T \)
61 \( 1 + (0.733 - 0.680i)T \)
67 \( 1 + (-0.826 + 0.563i)T \)
71 \( 1 + (0.0747 + 0.997i)T \)
73 \( 1 + (-0.988 - 0.149i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.955 - 0.294i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
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

−25.57177659743445916303155527253, −24.11442902898183518350309029436, −23.27896045858976187840604415826, −22.58085044900029810764633583543, −21.895882055890145124740292809629, −21.20192505839257753291378325558, −19.71044847990135036591660747099, −18.9172240850464788079540745830, −18.119717927968780008677931505093, −17.2696221402681839645624701334, −16.24999411670807045427887772235, −15.256744476478215208441637669205, −14.893540387709936411096906666591, −13.29948835229971593970191442387, −12.296453255288932466320939535907, −11.49172355552161252905698491837, −10.73483498826467013642655638511, −9.74261556118719283307081904081, −8.707207709174411367372445463600, −7.21345544013007645197023826426, −6.32858807796358817852403414522, −5.68407201589790635325372148778, −4.1469529727243964854049705166, −3.32800901213506639933488347361, −1.68074688672215772653263823684, 0.50462684506763882846723171351, 1.541325257640085020074839712, 3.702960490271451791942449829782, 4.43956725750664130838660527855, 5.71193184332105599075733370517, 6.57859771167358754607665728686, 7.59875056792098661192185874547, 8.75705568197014149162247652246, 9.83689730844157141333518021882, 11.00873563316927661473386582494, 11.67462004020968847847067057198, 12.78116786710555429975211719333, 13.3038392735551725521770015673, 14.52642153833313801435117673305, 16.1033112928671143631902740345, 16.510507216793872135373661683298, 17.05793467257369187699252098633, 18.33249945050033108162814023502, 19.13016564921489706322171880301, 20.14666327893238270611789024957, 20.92069493425684478492442158390, 22.065730234949091319871611534865, 23.08751402483622308653174727019, 23.41261731303729648648850901935, 24.47007879403657105201505748883

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